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Math In A Sustainable SocietyEdition 2.1

by Pete KaslikPierce College, Fort Steilacoom

Cover Photo

The photograph on the front cover was created by Chris Jordan (www.chrisjordan.com). The original photo measures 8 feet by 11 feet, in three vertical panels. “The photo depicts 2.4 million pieces of plastic, equal to the estimated number of pounds of plastic pollution that enter the world’s oceans every hour. All of the plastic in this image was collected from the Pacific Ocean.” (Jordan, 2009)

Below is a close up view that shows the details of Mt. Fuji.

Math In A Sustainable SocietyEdition 2.1

by Pete Kaslik

2010

Creative Commons License Attribution: Noncommercial 3.0 You are free to share or remix this work.

The edition number is written in the form M.m where M represents a major revision and m represents minor revisions, such as typographical errors or the addition or correction of problems. Minor corrections may result in some changes in page numbers compared to earlier editions, but these changes should be minimal.

http://creativecommons.org/licenses/by-nc/3.0/us/

To Jean

Acknowledgements

I am very grateful for the willingness of Chris Jordan to allow me to include one of his photos on the cover of this book. Issues of mass consumption result in numbers that are so big that people have difficulty comprehending them. By taking those large numbers and reducing them to a photo that represents only a small slice of time, the viewer is better able to grasp the magnitude of the numbers and consequently the magnitude of human actions.

I am very appreciative of the help, support and editing by my wife, Jean. This book has not had the benefit of professional editors so her feedback has been very valuable.

I appreciate my brother Jim Kaslik, for allowing the use of the picture of the dome home he designed.

I wish to acknowledge David Lippman, chairman of the Math department at Pierce College, for reviewing the book and for his support for my using it. I also wish to thank the classes of students on whom I experimented with earlier editions of this book. Their suggestions lead to the removal of some chapters that were less interesting and to the clarification of text, activities and homework that were confusing.

Table of Contents

Chapter 0.5 Quantitative Assessment of the World Activity....................................................1Chapter 1 Financial Survival.......................................................................................................5

SAVING IN ADVANCE...................................................................................................................6GROWTH RATES AND GROWTH FACTORS................................................................................9FUTURE VALUE WITH ARITHMETIC GROWTH.........................................................................9FUTURE VALUE WITH GEOMETRIC GROWTH........................................................................10THE COMPOUND INTEREST FORMULA....................................................................................11ANNUAL PERCENTAGE YIELD..................................................................................................12CONTINUOUS COMPOUNDING..................................................................................................14GRAPHING EXPONENTIAL GROWTH.......................................................................................15RULE OF 72................................................................................................................................20SINKING FUND............................................................................................................................23BIG PURCHASES..........................................................................................................................27

MONTHLY PAYMENT............................................................................................................27AMORTIZATION.....................................................................................................................28THE EFFECT OF PREPAYMENT.............................................................................................29

CREDIT TROUBLE.......................................................................................................................32Chapter 1.5 Sustainability..........................................................................................................41

If-Then Project.........................................................................................................................44System Dynamics Models........................................................................................................48

Chapter 2 Population Growth...................................................................................................53MODELING POPULATION GROWTH.........................................................................................56

Chapter 3 The Algebra of Sustainability..................................................................................62Chapter 4 Statistics.....................................................................................................................85

EXPERIMENTS AND STUDIES....................................................................................................88SAMPLING..................................................................................................................................89PROBABILITY............................................................................................................................93

SIMPLE PROBABILITY...........................................................................................................93P(A OR B)..............................................................................................................................94P(A AND B)............................................................................................................................95

USING DATA TO ANSWER QUESTIONS....................................................................................96GRAPHING QUANTITATIVE DATA............................................................................................96STATISTICS FOR QUANTITATIVE DATA...................................................................................98

STANDARD DEVIATION.......................................................................................................101THEORY...................................................................................................................................102SAMPLING DISTRIBUTION OF SAMPLE MEANS.....................................................................105CENTRAL LIMIT THEOREM....................................................................................................106CONFIDENCE INTERVALS.......................................................................................................106GRAPHS AND STATISTICS FOR QUALITATIVE (CATEGORICAL) DATA.................................111THEORY...................................................................................................................................112CONFIDENCE INTERVAL FOR PROPORTIONS........................................................................113

Chapter 5 System Dynamics Modeling...................................................................................123STOCKS, FLOWS AND CONVERTERS......................................................................................123

CAUSAL LOOP DIAGRAMS......................................................................................................126Computer Modeling...............................................................................................................128

Appendix.....................................................................................................................................134

1

Chapter 0.5 Quantitative Assessment of the World Activity

Many students, particularly those taking algebra, wonder when they will use math in their life. All too often, the justification involves things like giving change or balancing check books. These are small applications that require only arithmetic. In this book, you will get to see some larger applications of mathematics that will help you understand both personal and global issues and the decisions that can be made as a result of this understanding. The issues that will be addressed are those that cannot be understood without the mathematics.

To begin the process, you will look at a multitude of issues facing humanity. These issues are provided on the next two pages. Look at the issues presented and find one to three issues that interest you. In class, you will be able to sign up for one issue.

Your responsibility for this activity is to find one or two graphs that will help the class understand the issue. An ideal graph will show the status of the issue today as well as historically. In this context, today means during the last 1 to 5 years. Historically means over the last few decades. Projections are acceptable too. If a temporal graph is not available, then a spatial graph, such as one that shows the current status in the US and other countries should be used. It is critical that either a temporal or spatial comparison is made as numbers in isolation do not hold much meaning.

Ultimately, the class will watch the presentation and evaluate the topic on a scale of 0 to 4, in which 0 represents a critical state with a negative trend and 4 represents an excellent state with an improving trend. Consider a critical state as one that could negatively affect us during our lifetime. An excellent state is one that humanity should be proud of achieving.

Not all topics have the same importance. Besides scoring each topic, you will also give it a weight using numbers between 0 and 3. A score of 0 means you don’t consider the topic to have any importance at all to the well-being of life on earth. A score of 3 means you think the topic is extremely important to life. After viewing all graphs, you will find the weighted mean of your scores.

The graph should be copied into a Word document and sent to me as an email attachment. I will compile the graphs. Each graph must include the source (URL). Graphs are due to me by ______________.

This QAW project will be evaluated using the following criteria and points.1. Provides useful information so audience can make a reasonable judgment (10)

2. Includes current status (5)3. Includes temporal or spatial comparison (5)

4. Source (URL) provided with graph (2)5. Graph submitted on time (6)6. Presentation (competent and given when scheduled) (5)7. Watch presentations and judge graphs (2)

Total: 35 points

Math In A Sustainable Society 2.1

2

Topics Suggestions for information your graph should show

Human Health and Well Being1. Human Population WA Changes in total population over time. You can also

show changes in the ethnic composition.2. Human Population US Changes in total population over time. You can also

show changes in the ethnic composition.3. Human Population World Besides showing changes in total population, show

changes in first world, second world and third world countries.

4. Poverty US Show how the value that indicates poverty and the number of people in poverty has changed over time.

5. Poverty World Show poverty levels in first world, second world and third world countries.

6. Violent Crimes US Compare per capita violent crime rates in the US to other nations.

7. Hate Crimes US Compare per capita hate crime rates in the US over time or in various states.

8. War Deaths per year as a result of war or comparison of number of deaths in various wars dating back to at least the Civil War.

9. Prisons Show changes in prison populations. Show change in cost to government.

10. Life Expectancy Show life expectancy changes over time in the US. Compare with other countries too.

11. Health Care Cost Compare US to other nations12. Gender Relations Compare equality of men and women by showing a

comparison for salaries for the same job, students in higher education, proportion of women in positions of authority (management, government)

13. High School Graduation Rate Compare how graduation rates have changed over time.Food14. Marine Fisheries Show changes in size of fish stocks from around the

world.15. Farms Show changes in the number of farms.16. Farmers Markets Show changes in the number of farmers markets17. Water Quality Show changes in water quality18. Water Quantity Show major US aquifers and changes in water levels19. Bees Show changes in the honey bee population over the last

10 years.


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Environment and Energy20. US Oil Production and Consumption

Show US oil production – explain peak oilShow US oil consumption

21. World Oil Production and Consumption

Show world oil productionShow world oil consumption

22. Oil Costs Show costs per barrel and costs at the pump23. Natural Gas Production Show world production and changes over time.24. Coal Production Show world production and changes over time.25. Driving Distances Show the per capita annual distance driven in the US.

Compare to past years or to other countries.26. Air Pollution Show changes in carbon dioxide and at least one other

including methane, ozone, acid rain27. Climate Change Show global temperature changes.28. Electrical Energy Compare the amount of energy produced by various

sources such as fossil fuels, nuclear, hydro, wind etc. and show how that has changed over time.

Financial29. National Debt Show changes in National Debt from at least the early

part of the 1900s30. Housing Size and Occupants Show changes over time in the size and number of

people living in a house. Also, compare to other countries.

31. Housing Costs Show changes in the cost of housing in US and Washington

32. Wealth Gap Show the Gini Coefficient for the US and other countries. Explain the Gini Coefficient.

33. College Education Show how costs have changed over time.34. Per Capita Income Show how income has changed over time. Compare it

with inflation.35. Inflation Show US inflation rates over time.


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QAW Score CardName: Importance

Weight (0-3)Score (0-4) Multiply the Weight

times the Score

Human Health and Well Being 1. Human Population WA2. Human Population US3. Human Population World4. Poverty US5. Poverty World6. Violent Crimes US7. Hate Crimes US8. War9. Prisons10. Life Expectancy11. Health Care Costs12. Gender Relations13. High School Graduation RateFood14. Marine Fisheries15. Farms16. Farmers Markets17. Water Quality18. Water Quantity19. BeesEnvironment and Energy20. US Oil 21. World Oil 22. Oil Costs 23. Natural Gas Production24. Coal Production25. Driving Distances26. Air Pollution27. Climate Change28. Electrical EnergyFinancial29. National Debt30. Housing Size and Occupants31. Housing Costs32. Wealth Gap 33. College Education34. Per Capita Income35. InflationTotal ∑W = xxxxxxxxxxxxxxx ∑(W*S)=Weighted Mean xxxxxxxxx ∑(W*S)/∑W = xxxxxxxxxxxxxx


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Chapter 1 Financial Survival

We will start understanding the importance of math in society by looking at an issue of concern to most people. That issue is money. Understanding money is critical in the US consumer society for which the primary means of survival is based on the ability to purchase what is needed. Just because you were born into such a society and this society may be all you know, does not mean that it is the only way people can live. Some cultures, both historically and in current times, do not share our concern with the accumulation of wealth. However, since money is important for survival in our culture, our initial focus in this book will be on financial math, so that more prudent financial decisions can be made.

The economic recession that began at the end of 2007 and continued through 2009 has been the first major economic shock that many of those now living have faced. Lead by America, many parts of the world have seen virtually continuous economic growth since the Great Depression. Being born into a continually expanding economy has allowed most to accept, without question, that an economy will always grow and that whatever a person wants they will be able to buy, even if they need to borrow money to be able to afford it. Mortgages for big houses, loans for cars and college, shopping trips to the mall that can be charged to a credit card and pay day loans for dire situations are the key to keeping the economy growing. But the reality of the credit crunch of 2008 and the foreclosures of millions of homes may be that building an economy on credit is not sustainable and that a new model of finances may be required.

To help understand your financial needs and wants, use the table below to list what you own or rent or buy with your money, what is provided for you by others (e.g. parents), what you will buy in the future, and how much you expect it to cost.

Own/rent/ buy

Parents provide

Will buy in the future

Expected Cost

Food/water xxxxxxShelter (home/apartment)BicycleCarRecreation vehicle (boat, jet ski, RV, etc)TVComputerCell PhoneEducation


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For many of the items you listed, money is required. While most people get this money by having a job, a very few get it through inheritance or by winning the lottery. Assuming you are one of the multitudes that will have to work for your money, then you have three choices of how to get what you want.

1. Save in advance and then buy when you have enough money2. Use credit - buy now, pay later3. Do without or reduce the size of what you want

While many might be amazed at what they can live without, even with the awareness that they may already be living without it and that billions of other people in the world are living without it, we often believe our happiness is connected to having certain things. With this in mind, we will explore the mathematics of saving in advance and using credit, so that you will be able to make more informed decisions about getting the things you want.

SAVING IN ADVANCE

There are, in general, two ways for saving money for a future purchase. One way is to make a one time deposit of money into some sort of an investment option (stocks, bonds, mutual funds, savings accounts, money markets accounts, certificates of deposit) and let it increase in value until you need it. The second way is to put a little money into an investment on a regular basis and keep doing that until you need it.

The growth of that money can occur in two ways, it can be either arithmetic growth or geometric (exponential) growth. To understand the difference between the two, consider that you are offered a job and then given the choice of how you would like to be paid. The job will last for 30 days. Payment option 1 is to be paid $1000 dollars a day. Payment option 2 is to be paid 1 cent ($0.01) on the first day and then have your pay doubled every day. Which option would you choose? The table on the next page compares the two payment options.


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Option 1 Option 2Day Number Daily Cumulative Daily Cumulative

1 $1,000.00 $ 1,000.00 $ 0.01 $ 0.01 2 $1,000.00 $ 2,000.00 $ 0.02 $ 0.03 3 $1,000.00 $ 3,000.00 $ 0.04 $ 0.07 4 $1,000.00 $ 4,000.00 $ 0.08 $ 0.15 5 $1,000.00 $ 5,000.00 $ 0.16 $ 0.31 6 $1,000.00 $ 6,000.00 $ 0.32 $ 0.63 7 $1,000.00 $ 7,000.00 $ 0.64 $ 1.27 8 $1,000.00 $ 8,000.00 $ 1.28 $ 2.55 9 $1,000.00 $ 9,000.00 $ 2.56 $ 5.11

10 $1,000.00 $ 10,000.00 $ 5.12 $ 10.23 11 $1,000.00 $ 11,000.00 $ 10.24 $ 20.47 12 $1,000.00 $ 12,000.00 $ 20.48 $ 40.95 13 $1,000.00 $ 13,000.00 $ 40.96 $ 81.91 14 $1,000.00 $ 14,000.00 $ 81.92 $ 163.83 15 $1,000.00 $ 15,000.00 $ 163.84 $ 327.67 16 $1,000.00 $ 16,000.00 $ 327.68 $ 655.35 17 $1,000.00 $ 17,000.00 $ 655.36 $ 1,310.71 18 $1,000.00 $ 18,000.00 $ 1,310.72 $ 2,621.43 19 $1,000.00 $ 19,000.00 $ 2,621.44 $ 5,242.87 20 $1,000.00 $ 20,000.00 $ 5,242.88 $ 10,485.75 21 $1,000.00 $ 21,000.00 $ 10,485.76 $ 20,971.51 22 $1,000.00 $ 22,000.00 $ 20,971.52 $ 41,943.03 23 $1,000.00 $ 23,000.00 $ 41,943.04 $ 83,886.07 24 $1,000.00 $ 24,000.00 $ 83,886.08 $ 167,772.15 25 $1,000.00 $ 25,000.00 $ 167,772.16 $ 335,544.31 26 $1,000.00 $ 26,000.00 $ 335,544.32 $ 671,088.63 27 $1,000.00 $ 27,000.00 $ 671,088.64 $ 1,342,177.27 28 $1,000.00 $ 28,000.00 $1,342,177.28 $ 2,684,354.55 29 $1,000.00 $ 29,000.00 $2,684,354.56 $ 5,368,709.11 30 $1,000.00 $ 30,000.00 $5,368,709.12 $10,737,418.23

Option 1 represents arithmetic growth. It doesn’t matter how much money you have, your amount of money increases by the same amount with each time period. Option 2 represents geometric growth. The more money you have, the more money you get paid.

It is obvious that geometric growth ultimately results in the greatest increase. So what is geometric growth?

Consider the following two sequences of numbers:

Set A: 100, 150, 200, 250 …Set G: 100, 150, 225, 337.5 …For each set, the original number is n0, the next number is n1, the next is n2, etc.


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We will first consider the difference between two consecutive numbers in the sequence. This difference can be shown as nt+1 – nt where t is any value in the sequence. This means that if t = 0 then nt+1 – nt = n1– n0. If t=1 then nt+1 – nt = n2 – n1.

If we look at the difference between consecutive numbers in set A, we find the difference is always 50; for example, 150-100 = 50 and 200 – 150 = 50.

If we look at the difference between consecutive numbers in set G, we find the difference changes; for example, 150-100 = 50 and 225-150 = 75.

Because the difference between consecutive numbers in Set A is always the same, we conclude that Set A is showing arithmetic growth, but Set G isn’t, because the differences change.

Next, let’s look at the ratio of consecutive numbers.

For set A, the ratio when t = 0 is while the ratio when t = 1 is

.

For set G, the ratio when t = 0 is while the ratio when t = 1 is

.

Because the ratio between consecutive numbers is always the same in Set G, we conclude that set G is showing geometric growth. Set A is not showing geometric growth because the ratio changes.

From these examples we will conclude that for arithmetic growth, nt+1 – nt = a common difference. If it is money that is growing, the common difference is the amount of interest that is earned. Thus nt+1 – nt = I, where I is interest. With a little algebra, we can see that the amount at time t+1 equals the amount at time t plus the interest: nt+1 = nt + I

For geometric growth equals a common ratio, which we will call the growth factor.

For set G, the growth factor is 1.5. Using algebra, we can solve for n1 to get n1= n0·1.5.

Therefore, the amount at time t+1 equals the amount at time t times the growth factor.

GROWTH RATES AND GROWTH FACTORS


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Since the focus of this chapter is on money, then it is necessary to understand growth rates in terms of money.

We are accustomed to hearing rates on TV and radio advertisements. Car companies advertise car loan rates and mortgage companies advertise mortgage rates. These rates are growth rates which are typically given as a percentage. Percents are difficult to use in calculations, so the percent is typically converted to a decimal by dividing the percent by 100. For instance, an interest rate of 6% converted into a decimal is 0.06, which is a growth rate. The growth rate is represented with the variable r. Common names for r are the annual percentage rate (APR) and the nominal rate.

Multiplying the annual growth rate times the amount of money invested gives the amount of interest that is earned in 1 year. This can be shown with the formula I = P0r where P0 is the principal that is invested and r is the annual growth rate. The total amount of money in the account after one year is given by adding the principal that is invested and the interest that is earned. This is shown below, along with simplification by factoring.

Principal after 1 year = Principal + InterestP1= P0 + P0rP1= P0(1 + r)

where, P1 is the principal in the account after 1 year, P0 is the amount that was put into the account in the beginning (time 0), r is the annual growth rate 1+r is the growth factor.

If the interest rate is 6%, then the growth rate is r = 0.06 and the growth factor is 1.06. Thus, if you invest $100 in an account with 6% interest, after one year the account will have $106.

P1 = P0(1 + r)P1 = 100(1 + 0.06)P1 = 100(1.06)P1 = 106

Be aware that there are two different questions that could be asked. The first is about the amount of interest after a given amount of time (Pt-P0), and the second is about the amount in the account (Pt) after a given time. Most of the time the objective will be to find the amount in the account.

FUTURE VALUE WITH ARITHMETIC GROWTH

Planning for your future can be helped by anticipating the amount of money your investments will be worth at some future time. The future value of an investment is dependent upon the interest rate, the time and whether growth is arithmetic or geometric.

To account for times of more than 1 year, the interest formula during arithmetic growth is changed from I = P0r to I = P0rt. The formula for the amount of money after t years would


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change from P1 = P0(1 + r) to Pt = P0(1 + rt) For an investment of $100 at 6% for 3 years would result in interest of $18 and a total value of $118 as shown below.

I = P0rtI = 100(0.06)3I = 18

Pt = P0(1 + rt)P3 = 100(1 + (0.06)3)P3 = 100(1.18)P3 = 118

Arithmetic growth on investments occurs when the interest that is earned is removed from the account, so that the principal always remains the same. As will soon be evident, by keeping the interest in the account, growth can be geometric, which will ultimately result in more money.

FUTURE VALUE WITH GEOMETRIC GROWTH

For geometric growth, it is easier to find the amount in an account first and then use this to find the amount of interest that is earned.

If the interest rate is 6%, then the growth rate is r = 0.06 and the growth factor is 1.06. To find a way to determine the amount of money, we will use a geometric growth model. The original principal will be $100 so therefore P0 = $100. The value we will get after the first time period is P1 = P0(1+r) = 100 · 1.06 = 106 The value we get after the second time period is P2 = P1(1+r) = 106 · 1.06 = 112.36: Notice

that this can also be determined by substituting P1 = P0(1+r) into the equation P2 = P1(1+r) which will give P2 = P0(1+r)(1+r). Thus P2 = 100·1.06·1.06 = 112.36. The nice thing about this approach is that we can find the value after the second time period by knowing only the starting amount and the growth rate. We can simplify P2 = P0(1+r) (1+r) to P2 = P0(1+r)2.

In a similar way, P3 = P0(1+r)3 = 100·1.06·1.06·1.06 = 119.10 To be more general, Pt = P0(1+r)t Where

Pt is the value after t yearsP0 is the starting valuer is the annual interest ratet is the number of years for the investment

The formula Pt = P0(1+r)t is the simplified form of the compound interest formula. We will modify it shortly so it can be used in more cases, but first, we will try an example.

Example 1.1: Suppose you have $1000 in an account that pays 5% interest at the end of each year. Arithmetic growth occurs because the interest is not reinvested. Geometric growth occurs


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because the interest is reinvested. The table below shows a comparison of the interest earned each year as well as the total money at the end of four years.

Arithmetic Growth – don’t reinvest interest Geometric Growth – reinvest interestEnd of Year

Principal Rate Interest Total Principal Rate Interest Total

1 $1000 0.05 $50.00 $1050 $1000 0.05 $50.00 $10502 $1000 0.05 $50.00 $1100 $1050 0.05 $52.50 $1102.503 $1000 0.05 $50.00 $1150 $1102.50 0.05 $55.13 $1157.634 $1000 0.05 $50.00 $1200 $1157.63 0.05 $57.88 $1215.51

Notice how much more money there is with geometric growth than with arithmetic growth. The reason is that there is compound interest, that is, the interest is earned on the principal and accumulated interest rather than just on the principal. Compound interest occurs when the interest is left in the account.

THE COMPOUND INTEREST FORMULA

Interest is posted to the account at different times. Some financial institutions post it annually; others post it quarterly or monthly, while others may post it daily or continuously. To understand the difference between the various compounding periods, we must first determine the number of periods in a year. We will let this value be represented by k.

Compounding period Number of periods in a year (k)Annual 1Quarterly 4Monthly 12Daily 365 Continuously Infinite

The interest paid in each period is equal to the APR/k. For example, if the APR is 5% and it is compounded quarterly, then the quarterly interest rate is 0.05/4 = 0.0125.

$1000 invested at 5% compounded quarterly End of Quarter

Principal Rate Interest Total

1 $1000 0.0125 $12.50 $1012.502 $1012.50 0.0125 $12.66 $1025.163 $1025.16 0.0125 $12.81 $1037.974 (1 yr) $1037.97 0.0125 $12.97 $1050.955 $1050.95 0.0125 $13.14 $1064.086 $1064.08 0.0125 $13.30 $1077.387 $1077.38 0.0125 $13.47 $1090.858 (2 yrs) $1090.85 0.0125 $13.64 $1104.49

Notice that compounding quarterly results in more money than when compounding annually.


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We can modify the original compound interest formula to account for more frequent

compounding. The modified formula is Where

Pt is the value after t yearsP0 is the starting valuer is the annual interest rate (APR)t is the number of years for the entire investmentk is the number of compounding periods in a year

We will now use this formula on example 1.1.

ANNUAL PERCENTAGE YIELD

As can be seen in example 1.1, after one year, we have actually increased the value of the account by more than 5%. This is typical when interest is compounded. As already mentioned, the interest rate that is stated for an account is called the nominal rate or the Annual Percentage Rate (APR). The interest rate that is actually earned as a result of compounding is called the effective rate or Annual Percentage Yield (APY). Annual Percentage Yield can be found by calculating the rate portion of the compound interest equation for one year, then subtracting 1:

. In the example of 5% compounded quarterly, we get

.

You can verify this is the correct value by finding the actual percent increase after one year.

For each of the following compounding periods, find the amount of money in an account after 2 years if the initial principal (P0) is $4000 and the interest rate is 8%. Find the APY (rounded to 4 decimal places).

Annual Compounding


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Quarterly Compounding

Monthly Compounding

= = $4691.55

= = 0.0830

Daily Compounding

CONTINUOUS COMPOUNDING

Did you notice the APY increases with more frequent compounding, but that the increase is less each time? Supposing you compounded twice a day, or every minute or every second, would there be a limit to the increase in APY? It turns out that there is a limit. This limit occurs when you have continuous compounding. To understand this increase, consider the following


14

example. We will invest $1 in an account that pays 100% interest (r = 1.00) and use various compounding periods.

Annually = 2

Daily= 2.714516

Every Minute= 2.718283

Every Second= 2.718282

1 Billion Times a Year =2.718282

Notice how the value of Pt eventually stays the same, even as the compounding increases significantly. The value that is reached, 2.718282… is called e. The value e is used in cases that have continuous compounding. The formula for continuous compounding is: Pt = P0ert Where

Pt = the value of the account after t yearsP0 = the initial principale = 2.718282… although you should use the ex key on your calculator r = APRt = the number of years the money is invested

The APY when interest is compounded continuously is er-1.

Using our prior example of a $4000 investment at 8%, if the investment was compounded continuously, the value after two years would be:

Pt = P0ert

Pt = 4000e0.08*2

Pt = 4694.04

APY = er-1APY = e0.08-1APY = 0.0833

GRAPHING EXPONENTIAL GROWTH

In algebra you learned to graph equations on a Cartesian coordinate system graph. One of the graphing methods was to use a table of values. We will use this method to graph the compound interest formula equations. These are exponential equations because the independent variable t is in the exponent.


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To illustrate the graphing method, we will assume an interest rate of 3% and an initial investment of $1000. For one graph, we will use quarterly compounding, for another, we will use continuous compounding. To make the comparison more meaningful, we will also include a graph for no compounding, which is arithmetic growth. Its equation is linear. The variable t represents time, in years.

On the graph, notice that there is very little difference between the quarterly compounding and the continuous compounding, but notice that as time goes by, the difference between the compounding graphs and the arithmetic growth graph increases. The shape of the graphs for which there is compounding is typical for exponential functions.

Table of values for graphing geometric and arithmetic equations using the equations:

, Pt = 1000e0.03*t, Pt = 1000(1 + 0.03t)

t quarterly continuous no compounding0 1000.00 1000.00 1000.001 1030.34 1030.45 1030.002 1061.60 1061.84 1060.003 1093.81 1094.17 1090.004 1126.99 1127.50 1120.005 1161.18 1161.83 1150.0010 1348.35 1349.86 1300.0020 1818.04 1822.12 1600.0030 2451.36 2459.60 1900.0040 3305.28 3320.12 2200.0050 4456.67 4481.69 2500.0060 6009.15 6049.65 2800.0070 8102.43 8166.17 3100.0080 10,924.90 11,023.18 3400.00

This Page Is Available For Notes, Doodling, Ideas or Computations.


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In-class Activity 1.1: Using the Compound Interest Formula

Name _____________________________ Effort ___/3 Attendance ___/1 Total ___/4

Pick the correct formula, show the formula, substitution and solution.


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Pt = P0(1+r)t

Pt = P0ert APY = er-1

Your car currently has 130,000 miles on the odometer. You are hoping it will make it to 200,000, which means it will last approximately 7 more years, based on the average amount you drive each year. You don’t have car payments now, and would prefer not to have them in the future. You have $2500 that you would like to invest in a 3.1% certificate of deposit, compounded monthly. If you make this investment, how much money will be available for buying a new car in 7 years?

_______________________

What is the APY of this CD? ________________________



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We have now seen the normal ways that financial institutions offer compounding. We will use these methods to solve problems.

Example 1.2: An investor places $5000 in an account that pays 3.5% interest, compounded daily. How much money will the investor have in 10 years if there are no other deposits or withdrawals? What is the APY?


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Solution 1.2: Use the compound interest formula because there is daily compounding.

Example 1.3: An investor places $3000 in an account that pays 2.75% interest, compounded continuously. How much money will the investor have in 20 years if there are no other deposits or withdrawals? What is the APY?

Solution 1.3: Since there is continuous compounding, we need the continuous compounding formulas.

Pt = P0ert

Pt = 3000e0.0275*20

Pt = 5199.76

APY = er-1 APY = e0.0275-1 APY = 0.02788

Example 1.4: Planning ahead. Suppose the parents of a newborn want to have $20,000 in a college fund in 18 years. How much money must they invest now, as a one-time investment, to achieve their goal if the investment pays 4.6% interest, compounded monthly?

Solution 1.4: Because compounding is monthly, we need the compound interest formula. This time, however, we know the value after 18 years (Pt) but we don’t know the initial value (P0).


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$

$$

Example 1.5: Planning ahead. Suppose the parents of a newborn want to have $20,000 in a college fund in 18 years. How much money must they invest now, as a one-time investment, to achieve their goal if the investment pays 4.6% interest, compounded continuously?

Solution 1.5: Because compounding is continuously, we need the continuous compound interest formula. We know the value after 18 years (Pt) but we don’t know the initial value (P0).

Pt = P0ert

20,000 = P0e0.046*18

$$

RULE OF 72

There is a short cut way to estimate the growth of money. It is based on continuous compounding and will be explained without proof. Instead of calculating how much money will be in the account after time t, the short cut approximates how long it will take to double the initial principal. The banking industry uses the rule of 72. The rule of 72 says to divide 72 by 100*r. The result is the number of years it will take to double the initial principal. For example, if $3000 is invested at 2.75%, then the doubling period will be 72/2.75 = 26.2 years. We can check if this is approximately correct with the compound interest formula

Pt = P0ert

Pt = 3000e0.0275*26.2

Pt = 6166.38 – this result is slightly more than double our original investment of $3000.

To avoid confusion, please note that the Rule of 72 is the only formula in this chapter in which the interest rate is used as a percent rather than changed into a decimal.

In-Class Activity 1.2 : Using the Continuous Compound Interest Formula



Pt = P0(1+r)t


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Your car currently has 130,000 miles on the odometer. You are hoping it will make it to 200,000, which means it will last approximately 7 more years, based on the average amount you drive each year. You don’t have car payments now, and would prefer not to have them in the future. You have $2500 that you would like to invest in a 3.1% certificate of deposit, compounded continuously. If you make this investment, how much money will be available for buying a new car in 7 years?

_______________________

What is the APY of this CD? ________________________



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SINKING FUND

Up to now, we have considered one-time investments. That means we put the money into the account and leave it there to accumulate interest until the end of the time period. Most people do not have the necessary funds to make this type of investment. For most, the ideal is to invest a smaller amount of money each month. This is called a sinking fund. Just like problems with the compound interest, we would like to be able to calculate how much money we will have after time t if we make a regular monthly deposit and also how much our regular monthly deposit needs to be to achieve our goals. The formula for sinking fund is given without proof.


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where

Pt = the amount after time t (years)d = the regular depositr = APRk = the number of regular deposits per year

Most people make regular deposits once a month, so we will simplify the formula to solve that type of problem.

Example 1.6: A student decides to drink one less latte a day, thereby saving $2.50 per day. At the end of the month, the student has saved $75.00 ($2.50*30). Every month the student puts that money into an account that pays 4.5% interest. In 40 years when the student retires, how much money will be in the account? How much money will the student have put into the account?

Solution 1.6: Because we are using a regular monthly deposit, we need the sinking fund formula.

This is the amount in the account at the end of 40 years.

Since $75 is being deposited every month for forty years, the amount of money the

student put into the account is . By subtracting the

amount the student put into the account from the amount that was in the account after 40 years, (100,586.30 – 36,000) we find the student earned $64,586.30.


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In-Class Activity 1.3: Using the Sinking Fund Formula



Pt = P0(1+r)t



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Your car currently has 130,000 miles on the odometer. You are hoping it will make it to 200,000, which means it will last approximately 7 more years, based on the average amount you drive each year. You don’t have car payments now, and would prefer not to have them in the future. Your payment used to be $200 per month. You decide to pay that same amount each month to an account that will pay 3.1% interest. If you make this investment, how much money will be available for buying a new car in 7 years?

_______________________

How much of your money will you have put into the account? ________________________

How much interest would you have earned? ________________________


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BIG PURCHASES

Many students will eventually make big purchases in their lives. These include a home, a car, education, or a business. They are the types of purchases that generally require a loan. We will look at two aspects of loans, the first being to determine the monthly payment; the second is to understand how loans are paid off.

MONTHLY PAYMENT

If the interest rate on a loan remains fixed, then the amount of the monthly payment can be computed with the following monthly payment formula.

where:

P0 is the amount of the loanr is the APRt is the number of years of the loanM is the monthly payment

Example 1.7: What is the monthly payment of a 30-year, $120,000 mortgage with a 7% interest rate?

Solution 1.7:

M=$798.36. This is the monthly payment.If a person makes 360 monthly payments of $798.36, they will pay a total of $287,409.60

for the loan.The amount of interest they pay is $287,409.60 - $120,000 = $167,409.60

Example 1.8: Suppose the person had a 15-year mortgage instead of the 30-year mortgage. How much would the monthly payment be? How much would they pay for the loan? How much interest would they pay?


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Solution 1.8:

M=$1078.59

If a person makes 180 monthly payments of $1078.59, they will pay a total of $194,146.20 for the loan.

The amount of interest they pay is $194,146.20 - $120,000 = $74,146.20.

Notice that a shorter loan period means an increase in the monthly payment, but a decrease in the total amount of interest that is paid. When the house is paid in full, the person with the 15-year mortgage would have $93,263.40 more than the person with the 30-year mortgage ($167,409.60 - $74,146.20 = $93,263.40).

AMORTIZATION

It is important to understand what happens when you make monthly payments. Consider example 1.7 above. After the person signs the mortgage papers, they don’t owe the bank any money for one month. During that month, they have been borrowing $120,000. Because they are borrowing, the loan is accumulating interest and the bank wants to be paid that interest. The monthly payment will first be used to pay the interest, and then whatever remains will be used to reduce the principal. The best way to see this is with an amortization table.

Payment number

Interest Principal Balance

(interest/month *balance) Payment-interest Balance – Principal 0 $120,0001 $798.36-700 = $98.36 120,000-98.36 = $119,901.64

2 $798.36-699.43=$98.93 119,901.64-98.93 = $119,802.71

3 698.85 99.51 119,703.20An amortization table is constructed using 4 columns. The first column lists the months.

The second column lists the amount of interest that will be paid during the month. The third column lists the amount of the monthly payment that will be applied to the principal and the fourth column lists the new balance at the end of the month.


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When creating a table, start with month 0 to provide a row for the original balance (the amount that is borrowed and that must be repaid). The interest and principal columns in this row should remain empty.

In the interest column, calculate the amount of interest that is owed for that month. When learning about the compound interest formula you saw that the monthly interest rate was found by dividing the annual rate by 12 since there were 12 compounding periods in a month. A similar approach is taken to determine the amount of interest that must be paid during the month.

If the annual interest rate for the loan is 7.0% then the monthly interest rate is .

During the first month, you have been borrowing the original amount of the loan for one month. The institution that loaned you the money would like to be paid the interest. The amount of interest you owe at that time is the monthly interest rate times the original balance. In this example, we find the interest that is owed for borrowing $120,000 for one month at 7% interest

rate is $700.00. This is shown by the calculation .

Since the monthly payment of $798.36 exceeds the amount of interest, then the difference between the two is applied to the principal thereby reducing the balance. The difference is found by subtracting the interest from the monthly payment.$798.36 - $700.00 = 98.36.

In the last column, we see that the balance is reduced by the amount of principal that was paid. Therefore the new balance can be found by subtracting the value in the principal column from the balance of the previous month, 120,000-98.36 = 119,901.64.

Now the process starts all over again. For the next month, you will only be borrowing $119,901.64 rather than $120,000. In the interest column, multiple the new balance by the monthly interest rate. Notice that the amount of interest that must be paid is less than it was during the first month. In the principal column, subtract the interest for the month from the monthly payment to determine the amount that will be paid towards the principal. Notice that the amount paid towards the principal is slightly higher than in the previous month. Finally in the last column, find the new balance at the end of the month by subtracting the principal from the previous balance.

THE EFFECT OF PREPAYMENT

It is possible to pay more than your monthly payment. One way of doing this is to include the following month’s principal amount with the current month’s payment. For example, if the first month’s check was increased by 98.93 to $897.29, then you would save yourself $699.43, which is the amount of the interest you pay in the second month. You would not notice this savings until the loan is paid off however. By paying the principal for one month, you would actually finish paying for your loan one month early. Thus, instead of making 360 payments, you would only need to make 359. Paying the next month’s principal in addition to your regular payment will not allow you to skip the payment next month; it will only let you finish paying for the loan one month early.


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Prepaying the next month’s principal makes more sense early in the loan than later in the loan. Early in the loan, the amount that is applied towards the principal is a small amount, whereas it is much larger later in the loan. Following is the last few months for the amortization table. Prepayment at the end of the loan would require increasing your monthly payment $793.73 and the savings by not having to pay interest would be only $4.63. Thus, the prepayment benefits are greater when the prepayment is made early.

Payment number

Interest Principal Balance

(interest/month *balance) Payment-interest Balance - Principal358 $ 13.81 $784.55 $ 1,582.86 359 $ 9.23 $789.13 $ 793.73 360 $ 4.63 $793.73 $ 0.00

When borrowing money, take the time to read and understand the loan papers you are signing. One of the conditions within the loan papers that you should identify is that prepayments can be made at any time, without penalty. This way, you can reduce your debt quicker without being penalized.

In-Class Activity 1.4: Monthly Payments and Amortization



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Pt = P0(1+r)t


Your car finally reached 200,000 miles and you decided it was time for a new car. The new hydrogen fuel cell car you want will cost $38,000. Based on the combination of Activity 2 and 3, you have saved 3105.86 + 18,735.22 = 21,841.08. This money will be used for the down payment but you will need a loan for the balance. The best loan rate you can find is 8.6% for a 5 year loan.

Calculate your monthly payment.

_______________________

Complete the first 3 months of the amortization table.

Payment number Interest Principal Balance

Payment-interest Balance - Principal

0 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx

1

2

3

Notice how the amount of interest paid each month is gradually decreasing, the amount of principal paid each month is gradually increasing and the balance is gradually declining. After 60 months, the balance will be zero.CREDIT TROUBLE

As is evident during the economic downturn of 2008-2009, credit can cause problems for people. Borrowing beyond your means or allowing debt to accumulate on credit cards puts people into a difficult situation when jobs are cut or anticipated raises don’t materialize because


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the economy is depressed. Adopting some personal rules about how you will use a credit card can make your life less stressful. Some rules to consider about credit cards and other credit:

1. Pay the entire amount on the credit card each month, that way you won’t owe any interest and you will build your credit score for future borrowing.

2. Pay all loans on time so you don’t incur late fees.3. If you can’t afford a purchase, buy it later when you have saved enough money.4. Keep a low limit on your credit card.5. Have only one credit card and use it only for emergencies.6. If your credit card debt is growing, pay more than the minimum amount.7. Live simply – quality of life is not determined by how many things you have.8. If all else fails, cut up your credit cards and contact a credit counselor.

In times of desperation, some people resort to pay-day loans. These are loans that allow short term borrowing with expected payback periods of two weeks. For the convenience they offer, the borrower pays a high interest rate. Payday loans will be explored in the next activity. The interest rates that you will determine are realistic.

In-Class Activity 1.5: Payday LoansName______________________Points ___/4 Attendance ___/1 Total ___/5


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Your savings are gone, your checking account is nearly gone and you have a bill that must be paid. What do you do? For some, the solution is payday loans. Payday loans are short term loans that must be paid back when you get your next pay check. If they aren’t paid back, other fees will have to be paid.

One payday loan business lends money to customers with jobs. They charge $18 for each $100 that is loaned. The term of the loan is 14 days.

(1) 1. What is the amount of interest that must be paid for borrowing $500?

(1) 2. What is the interest rate, r? This is not the APR!

(1) 3. Since the term is k = 14 days, then we can calculate the daily interest rate by dividing r

by k, . What is the daily interest rate?

(1) 4. You can use the daily interest rate to determine the annual interest rate. That is done by multiplying the daily interest rate by 365. If you used the decimal form of the interest rate, then multiply that answer by 100 to find the interest rate as a percent. What is the annual interest rate (as a percent)?


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In-Class Activity 1.6: Excel Monthly Payment and Amortization Schedule Name________________________________ Points ______/15


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The goal of this activity is to use Microsoft Excel to find the monthly payment for a loan and then create an amortization schedule. The spreadsheet should be versatile in that you should be able to change input variables and have the spreadsheet recalculate. An example is shown below.

A B C D1 Cost2 Down payment3 Balance =B1-B24 Interest (as a decimal)5 Term (months)67 Monthly Pmt =PMT(B4/12,B5,-B3,0,0)8910 Period Interest Principal Balance11 =B312 1 =$B$4/12*D11 =$B$7-B12 =D11-C1213 2 =$B$4/12*D12 =$B$7-B13 =D12-C13The monthly payment formula is =PMT(rate as a decimal, number of payment period, Present Value (negative loan amount), Future Value (0), Payment at end of period (0))

You have decided to buy a house. The house will cost $169,000. You have saved enough for a down payment of $30,000. The interest rate for the mortgage is 5.9%, regardless of the term. You aren’t sure if you want a 15-year mortgage or a 30-year mortgage. To decide, you must consider whether the payments are affordable. A monthly mortgage payment should be less than 25% of your monthly income.

(1) 1. What is the monthly payment for the 15-year mortgage?(2) 2. What is the total amount of interest you will pay over the life of the mortgage?(1) 3. What is the monthly payment for the 30-year mortgage?(2) 4. What is the total amount of interest you will pay over the life of the mortgage?

(2) 5. What is the difference in the amount you will pay in interest over the life of your mortgage between the 30-year and 15-year mortgages?

(1) 6. If your monthly income is $3,800, which mortgage can you have so that your monthly payments are less than 25% of your income?

Select all that apply by underlining: 15-year 30-year

(6) 7. Complete the table below that shows the payment number, interest, principal and balance for the 180th payment of both the 15 and 30 year mortgages.

Payment number Interest Principal Balance15 year mortgage 18030 year mortgage 180


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Chapter 1 HomeworkName ____________________________________ Points _______/_____


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For questions 1 – 5 answer all of the following questions. How much money will the student have at the end of 5 years? How much interest will the student have earned in 5 years? Is this an example of arithmetic or geometric growth? What is the effective yield (APY)?

1. A student puts $1000 in a savings account that pays 2% annual interest. The interest is paid to the customer at the end of each year and is not reinvested.

2. A student puts $1000 in a savings account that pays 2% annual interest. The interest is reinvested.

3. A student puts $1000 in a savings account that pays 2% annual interest, compounded quarterly. The interest is reinvested.

4. A student puts $1000 in a savings account that pays 2% annual interest, compounded monthly. The interest is reinvested.

5. A student puts $1000 in a savings account that pays 2% annual interest, compounded daily. The interest is reinvested.

6. Use the results of problems 1 to 5 to make a graph of the number of compounding periods in a year and the APY. The number of compounding periods should go on the x-axis, the APY goes on the y-axis. Pick an appropriate scale.


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7. How much money do you have to invest one time in a 6% account compounded quarterly to have $10,000 in 4 years?

8. Suppose you have a one-year old child and want to invest some money for a college fund. You expect to need the money in 17 years. If you want to have $20,000, how much money will you need to put into the account if it pays 5%, compounded daily?

9. A student puts $1000 in a savings account that pays 2% annual interest, compounded continuously. How much money will the customer have at the end of 5 years? How much interest will the customer have earned in 5 years? Is this an example of arithmetic or geometric growth? What is the effective yield (APY)? Look at the graph in problem 6, does your answer make sense?

10. Use the rule of 72 to estimate the time it takes to double your principal if you invest at the following interest rates.

a. 12%


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b. 8%c. 6%d. 4%e. 2%f. 1%

11. A student estimates she needs $10,000 in 10 years. If she makes a one-time deposit, how much money must she put into an account that pays 6% annual interest, compounded continuously? How much interest will the customer have earned in 10 years? What is the effective yield (APY)?

12. If you deposit $50 per month for the next 15 years into a 5% account that is compounded monthly, what is the total amount of money you will have 15 years from now? How much interest will you earn?

13. If you deposit $25 per month for the next 8 years into a 4% account that is compounded monthly, what is the total amount of money you will have 8 years from now? How much interest will you earn?

14. How much money must you deposit per month to have $13,000 in 4 years if the APR is 3%, compounded monthly?

15. You need to borrow $8,000 to start a business. The bank offers a loan rate of 11% APR for a 6-year loan. What is your monthly payment? What is the total amount of money you will pay over 6 years if you don’t prepay?


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16. Complete the first 3 months of an amortization table for problem 15.

Month Interest Principal BalanceX X X X X X X X X X X X X X X X X X X X X X X $8000123

17. You borrow $14,000 for a new car. The bank offers a loan rate of 9% APR for a 5-year loan. What is your monthly payment? What is the total amount of money you will pay over 5 years if you don’t prepay?

18. Complete the first 3 months of an amortization table for problem 17.

Month Interest Principal BalanceX X X X X X X X X X X X X X X X X X X X X X X $14,000123

Chapter 1.5 Sustainability

The finance math that you learned in Chapter 1 is of relevance today, but whether that will be the case in the future may be determined by choices humans make. You should have just


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completed your first project called the Quantitative Assessment of the World. In looking at the various graphs provided by your classmates, there are a few things that you may have noticed.

1. The population of the world continues to grow (currently about 6.8 billion).2. The world’s ability to produce oil has probably peaked or will peak within your lifetime while demand will grow.3. The marine fisheries, a major source of food, are declining to the point of being critical. 4. Manmade toxic chemicals are found in our air, water and food.5. Global climate change threatens the planet.

This situation can be viewed in a least two different ways. The first way is the way with which you are most familiar. During your lifetime, you have had a nearly endless supply of anything you want. The food shelves at the local grocery stores have always been full with food produced thousands of miles away. The clothing stores have always had the latest fashions, with the clothing produced around the world. For many students, there is no memory of a time when there wasn’t a computer, cell phones or the internet. There has always been gas for your car. You have been told to learn as much as possible in school so you can get a job that will keep the society going and allow you to become a consumer in it. You are taught in schools so that you can use your knowledge to help solve any problem you encounter. With science fiction as inspiration, we know that we can solve all of our problems and eventually find other planets to inhabit, when there are too many on this planet.

A different way of viewing our world is from a long-term perspective. While humans have been around for over 4 million years, their numbers were small, relative to the size of the planet. The world population didn’t reach the two billion mark until the late 1920s. At that time, the US produced about 1.7 million barrels of oil per day. Currently, with 6.8 billion people, the world is producing about 86 million barrels of oil per day. By some estimates, the world cannot produce more than that amount ever again and yet there will be more people with more demand.

From the long-term perspective, consider the changes that have occurred in Washington State. In less than 200 years, Washington State has transitioned from a state with scattered Indian tribes and abundant natural resources to a state with about 7 million people, large homes and a massive, congested highway system. Streams once had millions of salmon return to spawn. Now, considerable human effort is required to keep some runs from becoming extinct. There are more people with more needs, but with fewer resources.

While the current financial problems can be attributed to anything from corporate greed and inadequate government oversight to too much government, consider for a moment, how much of our economic growth is connected to the availability of cheap energy. Our ability to extract resources, manufacture products, transport goods and people, construct buildings, grow and transport food and heat our homes, all require energy. As the population grows, the demand for energy grows, but Earth’s store of fossil fuels declines. This connection between our energy needs and our economic viability shows how fragile our economy is.

Consider the comments of Charles Galton Darwin (grandson of Charles Darwin) and those of Sir Fred.


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The fifth revolution will come when we have spent the stores of coal and oil that have been accumulating in the earth during hundreds of millions of years. … It is to be hoped that before then other sources of energy will have been developed, … but without considering the detail [here] it is obvious that there will be a very great difference in ways of life. … Whether a convenient substitute for the present fuels is found or not, there can be no doubt that there will have to be a great change in ways of life. This change may justly be called a revolution, but it differs from all the preceding ones in that there is no likelihood of its leading to increases of population, but even perhaps to the reverse. (Darwin, 1953, cited in Duncan, 2009).

Sir Fred Hoyle (astrophysicist) It has often been said that, if the human species fails to make a go of it here on the Earth, some other species will take over the running. In the sense of developing intelligence this is not correct. We have or soon will have, exhausted the necessary physical prerequisites so far as this planet is concerned. With coal gone, oil gone, high-grade metallic ores gone, no species however competent can make the long climb from primitive conditions to high-level technology. This is a one-shot affair. If we fail, this planetary system fails so far as intelligence is concerned. The same will be true of other planetary systems. On each of them there will be one chance, and one chance only. (Hoyle, 1964, cited in Duncan, 2009)

Is our future the one modeled by the TV show Star Trek or is it one defined by Peak Oil Theory and the Olduvai Theory which states that that the life expectancy of Industrial Civilization is less than or equal to 100 years (Duncan, 2009). Can we sit back without worry because the evidence of our lifetime is that the political leaders, scientists and engineers will continue to make life better for us as they have done during our lifetime or should we consider that infinite growth in a finite world is not possible and that an alternate strategy is needed for our benefit and the benefit of our children.

The premise upon which the rest of this course will be taught is that we are capable of developing a society that makes use of the knowledge we have gained while simultaneously eliminating our dependence on resources that cannot be replaced. We will assume humans are intelligent enough to continue to gain knowledge and create a world of equality and justice while not simultaneously destroying the means for future generations to do the same thing. Such a society would be sustainable. Sustainability means living in a way that life can be fulfilling, but without impacting the ability of future generations to have worthwhile lives too. Sustainability has three interrelated components- environmental, economical and social justice.

When thinking about changes to our society, keep in mind the words of Wade Davis, an anthropologist and author of “The Wayfinder, Why Ancient Wisdom Matters in the Modern World. In a podcast available through The Long Now Foundation, Davis said “…the other peoples of the world are not failed attempts at being us; they are unique answers to this fundamental question. What does it mean to be human and alive?” Any alternate vision for our future would simply be another unique answer.


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To help you envision an answer that includes sustainability, we will engage in a thought experiment. We will pretend that a group of concerned professors wish to determine how people can live sustainably without regressing to the Stone Age. Their intent is to create an experimental community which is isolated from the rest of civilization following its creation, with the exception of periodic visits from the professors to gather information that can be used as the world transitions to a sustainable society. The primary motivation for this particular thought experiment is that the numbers we encounter in trying to understand our world are so big that most people cannot comprehend what they mean. For example, how much is 86 million barrels of oil per day? What enormous effort, consumption of energy and cost would it take to replace the estimated 600 million gas powered cars that are in existence today with electric cars or hydrogen fuel cell vehicles? While we see these numbers and know what they represent, we cannot truly grasp them and therefore tend to ignore them.

Consequently, the community in our thought experiment will be small. It will start with 1000 people and a few assumptions.

1. There is no oil or other hydrocarbons.2. There is no commerce outside of the community.3. There is limited communication with those outside of the community. This means no

TVs, radios, internet.4. People will need to produce what they need with the resources that are available.5. What people destroy or consume will not be repaired or replenished, unless nature

does it.6. Of the 80 square kilometers of land available, only 20% may be altered by the

residents. This will help residents understand that humans are part of the world, not the only reason for the world to exist..

7. The primary objective of the community is to determine which portions of modern life we can use sustainably and which we can do without. We will assume that people need shelter, food and water. To be happy, they also need an opportunity to create meaning in their own life through learning, through experiences, or by creating art or music.

For the purpose of this thought-experiment, this community will be located in the Pacific Northwest. It will be on the coast, but will be surrounded by mountains ranging from 2500 to 4000 meters high. Its name will be Steilacoom Valley (In honor of Pierce College, Fort Steilacoom Campus, Lakewood, WA).

Steilacoom Valley will be used for learning the mathematics of sustainable living on a small scale, after which these skills can be applied to the large scale of our current world.

Consequences Project

Having recently lived through the highest gas prices ever, the worst recession since the Great Depression and the collapse of the housing market with lots of foreclosures, you should be keenly aware that world events have an impact on your life. The Quantitative Assessment of the


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World activity should have made you even more aware that there are a lot of issues that face all inhabitants of this planet. You have a choice of ignoring all of this information or you can become aware of issues and let them influence your choices.

The ability to think critically requires the formation of a question, the gathering of information and extensive time for reflection, during which various possibilities can be mentally explored. This project is designed to foster critical thinking that is grounded quantitatively, but is relevant to the world as it exists today (this has nothing to do with Steilacoom Valley). The prompt you will use is in two parts. The first part begins with the word “if” and states a condition that you will assume to be true. The second part starts with the word “then” and provides the topic for which you will make a hypothesis about a possible consequence. If possible, make a hypothesis that reflects a positive vision of the future.

Example: If mankind does not find a miracle source of energy for the future, then one consequence to our industrial/consumer way of life, once we are past peak oil production, is…

Some possible responses:… materialism will decrease (because of a decrease in manufactured products)… waste and pollution will decrease (because of a decrease in manufacturing and the products)… greenhouse gas production will decrease (because of a decrease in manufacturing)

Guidelines1. Form a group of 2 or 3 students.2. State the complete question with your hypothesis appearing after “then”. Your hypothesis should consist of one important expectation that you think will occur.3. Create a systems dynamics model (optional)4. Show relevant statistics and graphs that provide the background for the “if” portion of the question. Identify all sources on each slide.5. List your assumptions about the future. 6. Use original mathematical calculations to support your hypothesis. These can be formulas, dimensional analysis, graphs that you create, etc that support your hypothesis. Do not put in math, statistics or graphs that someone else created.7. Conclusion – connect all the pieces (background, assumptions and math) so they support your hypothesis.

Your project will be presented to the class in a 5-10 minute power point presentation. Create a one page outline that shows the question and your hypothesis, the systems dynamic model (optional), the background, assumptions and math. Submit both the outline and a copy of the power point electronically as attachments to an email. The outline is due by ____________ and the power point is due by _________________. Presentations will begin _____________.

Topic Prompts

1. If the world’s demand for petroleum products continues at the average rate of the last decade, but world production remains unchanged, then the consequences to economic growth are


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2. If China’s demand for petroleum products increases at the average rate of the last decade and their needs are met first because of political alliances with oil producing nations and if worldwide oil production remains unchanged, then the consequences to the US are3. If our country stops importing petroleum over the next 10 years, then the consequences to industry are4. If our country stops importing petroleum over the next 10 years, then the consequences to transportation are5. If our country stops importing petroleum over the next 10 years, then the consequences to electrical energy production and our homes are6. If our consumption of water continues at the average rate of the last decade, then the consequences to the world supply of fresh water are7. If petroleum and natural gas is unavailable for herbicides and fertilizers, food transport and farm machinery, then the consequences to the food supply are8. If the Columbia Ice Fields melt, then the consequences to farming, hydro power, fish and life in the Pacific Northwest are9. If petroleum is no longer available or is very expensive, then the consequences to living as a result of the decrease in plastic products are10. If the US population continues to grow at its current rate, then the consequences to our quality of life are11. If all aquifers continue to change at the rate of the last decade, then the consequences to the amount of farmland that can be watered is12. If ocean fishing continues at the rate of the last decade, then the consequences to marine fisheries are13. If the Federal Government was forced to eliminate the national debt in 10 years, using a constant annual rate of change (so they can’t put it off until the 10th year), then the consequences to federal programs are14. If all non-essential manufacturing (toys, furnishings, new construction materials, etc) was converted to the production of solar panels, wind mills and similar devices, then the consequences to our ability to live without oil are15. If the amount of energy used by each country must always be proportional to the population, then the consequences to this country would be16. If the amount of food consumed by each country must always be proportional to the population, then the consequences to this country would be17. If gasoline powered cars were not allowed and people put half of their annual car expenses into a mass transit system each year, then the consequences to the ability of people to get around would be18. If the US eliminated all oil imports in the next five years, then the consequences to the percent in poverty are

Sample Outline for Project


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Question: If mankind does not find a miracle source of energy for the future, then one consequence to our industrial/consumer way of life once we are past peak oil production is…

…materialism will decrease. Systems Dynamics Graph

Background: 1. Peak Oil2. Energy used for manufacturing3. EROI (Energy Return on Investment)

Assumptions: 1. The population will continue to grow2. People will continue to strive for a higher standard of living3. Unemployed people cannot purchase many products

Mathematical Argument:1. Show a regression of oil costs and unemployment.(lag 2 years).

Conclusion:1. The mathematical argument shows that as the cost of oil increases, so does unemployment. 2. World population is growing and many people want a higher standard of living.2. EROI on oil is declining and it is still low for non-renewable energy sources. 3. Since energy costs will climb, then it follows from the causal feedback loops that manufacturing, employment and sales will all decline, thus leading people to have fewer material possessions.

Evaluation

Outline (7), Power Point (7) and Presentation (6) on Time ___/20


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Power Point Excellent (9-10), Good (8) Fair (7) Poor (<7) ___/10

Speaking Excellent (9-10), Good (8) Fair (7) Poor (<7) ___/10

HistoryQuality (relevant and must contain sources)

Excellent (18-20), Good (16-17) Fair (14-15) Poor (<14) ___/20

AssumptionsQuality Excellent (9-10), Good (8) Fair (7) Poor (<7) ___/10

MathQuality Excellent (18-20), Good (16-17) Fair (14-15) Poor (14)

___/20

HypothesisSupport Excellent (9-10), Good (8) Fair (7) Poor (<7) ___/10

Total ____/100

Suggestions:

The only slides on the power point with a significant amount of text should be the hypothesis slide, the assumptions slide and the conclusion. Don’t use too much text or annoying text animation.

Stand in front of the class and talk about the information, don’t sit and don’t read it. Provide sources for all graphs and data. When presenting numbers in a table you make, round the numbers. For example, if you

find a population figure of 6,823,452,781, it is acceptable to round it to 6.8 billion. If you are discussing millions or billions of dollars in money, anything less than $1000 is trivial. The math must be of your own creation. Don’t put in graphs from another source.

Submit on time. 2 points deducted for each day it is late. 2 points deducted for each upgrade of your presentation after the time it was first due.

All group members need to talk during the presentation. Be highly selective of the slides you put in your presentation.

o 1 title slideo 1 hypothesis slide (text)o 3-6 background slides (graphs/tables)o 1 assumption slide (text)o 1-4 mathematics slides (math or graphs)o 1 conclusion slide (text)

System Dynamics Models


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It is typical that the analyses of problems faced by humans are viewed in isolation. However, topics such as climate change, population growth, peak oil, social justice and national debt are all connected. Solving these problems requires the ability to see the connections and understand how a change in one area can cause a change in another. In the 1960s, the field of System Dynamics was created by Jay Forrester and his colleagues at the Sloan School of Management at the Massachusetts Institute of Technology. This field allows researchers to model behaviors in a complex system then use computer simulation to explore impacts to the system when changes are made.

This section is going to provide only the most basic introduction to the concepts. Using these concepts, even at an elementary level, can significantly improve your critical thinking processes and will help inform the argument that you will present in your Consequences project.

The system will be viewed using stock and flow models and the causal loop diagram. A demonstration will be made with a model for a recent high school graduate considering options for the future. Stocks represent accumulation or storage in a system. For a student, this could be viewed as credits earned for a degree. Stocks are represented with rectangles or clouds. A cloud is a stock that exists without explanation and usually precedes or follows the stocks in rectangles. Assume the student wants to get an Associates degree at a community college. In this model, the cloud represents prior education such as high school, home school or GED, it is not important where the education was earned as far as this model is concerned. The cloud on the right represents what happens after the time at the community college (e.g. university, job, graduate school, Peace Corp), which is not of particular importance in this model.

After creating the stocks, it is very helpful to identify the units that will be measured. While it is tempting to use units of credit hours earned, it might be more interesting to observe the time spent at each stock.

Flows are actions that cause changes to the stock. There are two primary flows that will impact the time spent in school, student responsibilities and other activities. Student responsibilities include attending and participating in class, reading, doing homework and thinking about the concepts. Other activities include being an athlete, a parent, in the military or having to work.


Community College

Community College(time)

student responsibilities


other activities

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We would expect that both attending class and studying would help reduce the time spent in college, but life is often more complicated than this. For example, a student may have other expectations for their time because of family or work. To understand the impact of these, we will make causal loop diagrams. These diagrams show how one thing can affect another because of feedback. Feedback can be positive or negative. Positive feedback means that an increase in one variable leads to an increase in the next or a decrease in one leads to a decrease in the next. Negative feedback means that an increase in one variable leads to a decrease in the next, or visa versa.

We now need to assign a polarity to each causal arrow based on whether a change in one variable has a positive or negative effect on the next variable.

We would expect that the more responsible a student is because they attend class and do their homework, the less time the students would be at the college (because they would pass the courses the first time with a sufficiently high grade). This would have a negative polarity. We would also expect that the longer a student is in school, the more responsible the student becomes, because they learn how to be better students.

On the other hand, the more activities (job, being a parent) that a person has, the longer it will take to complete school (positive polarity). The longer a student takes to get through school, the more likely the student is to have other activities (job, parenthood, etc).


student responsibilities


other activities

Time in College student responsibilitiesother activities

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When there are an odd number of negative signs around the loop, the diagram shows a negative feedback loop. If there is an even number of negative signs, the diagram shows a positive feedback loop. Since there are no negative signs in the other-activities loop, then this is a positive feedback loop. This indicates that the more activities a student has, the longer it is likely to take them to get through college. The student-responsibility loop has an odd number of negative signs (1) so this is a negative feedback loop. This indicates that the more responsible a student is, the less time it will take them to get through school. Neither of these results should be a surprise.

Activity 1.5.1: System Dynamics ModelName _______________________________ Points ___/16 Attendance ___/3 Total ___/19


+ -+

Time in College student responsibilitiesother activities

+ -+

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In this activity, you will label the system that models food in your house. The stock represents the amount of food stores in your house. Let the units for the food be kilograms. The flows are the food that comes into the house and the food that is consumed. Let time have units of weeks. The flows will have units of kilograms/week.

(3) 1. Label the flow model with the appropriate stocks and flows. You do not need to label the clouds.

The following questions will help you determine polarity in the causal feedback loops

(1) 2. As the amount of food purchased each week increases, will the amount of food stored increase or decrease?

(1) 3. For the amount of food stored each week to increase, will the amount of food purchased have to increase or decrease?

(1) 4. As the amount of food consumed each week increases, will the amount of food stored increase or decrease?

(1) 5. As the amount of food stored each week decreases, will the amount of food consumed each week increase or decrease? Think what you would do if you couldn’t buy more food and your pantry was starting to get low.

(9) 2. Use the results of questions 2-5 to help you fill in the blanks in the Causal Loop Diagram below, then show the polarity of each arrow and indicate whether each loop is a positive or negative feedback loop using a large plus or minus sign in each loop.


__________ ________________________

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Chapter 2 Population Growth


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With the number of people putting a tremendous pressure on the world resources, it is useful to understand the mathematics of population growth.

There are four different factors that affect populations. These factors are the birth rates, the death rates, immigration and emigration. The last two are known collectively as migration. For a completely enclosed system such as the Earth, there is no migration, so only birth and death rates are relevant. For smaller regions, such as communities or countries, migration must be included.

Birth rates are the number of births per 1000 individuals. Likewise, death rates are the number of deaths per 1000 people. A population that has 20 births for every 1000 people has a

birth rate of or 2%.

The growth rate of any location is defined by:

Growth rate = birth rate – death rate + migration

If migration is a positive number, more people are immigrating than emigrating. Conversely, if migration is a negative number, more people are leaving the country than arriving.

All the following rates are based on information from the 2009 CIA – World Fact Book (CIA, 2009). Birth rates of different countries vary from a high of 51.6 per 1000 in Niger to a low of 7.42 per 1000 in Hong Kong. The US birth rate is 13.82, while the world’s birth rate is 20.18.

Death rates vary from a high of 30.83 per 1000 in Swaziland to a low of 2.11 per 1000 in the United Arab Emirates. The US death rate is 8.38 per 1000, while the world’s death rate is 8.23 per 1000.

Migration rates vary from a high of 22.98 in the United Arab Emirates to a low of -21.03 in Federated States of Micronesia. Migration in the United States is 4.31 per 1000, ranking it 25th in the world.

For the United States, the growth rate is 13.82 – 8.38 + 4.31 = 9.75 per 1000.

Another way to understand population growth is with the Total Fertility Rate (TFR). This is the average number of children born to a woman during her lifetime. It is actually a composite amount based on the expected number of births for women of different ages. TFR vary from a high of 7.75 children per woman in Niger to a low of 0.91 in Macau, which is similar to Hong Kong in that it is part of China, but has its own set of laws. The TFR in the United States is 2.05 and in the world it is 2.61.

Example 2.1: Suppose that the Steilacoom Valley population had similar birth and death rates as the United States. How many people will be in Steilacoom Valley in 12 years?


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Solution 2.1: This may, or may not be a realistic assumption, but we will make it to help explain the change in the Valley’s population. We will also assume immigration equals emigration, so that if anyone wants to join the Steilacoom Valley experiment, they could only do so in replacement of someone who wants to leave.

The first difficulty encountered with trying to use the US birth and death rates is rounding. A rate such as 13.8 births per 1000 people can make sense if there are millions of people, but if there are only 1000 people, then there cannot be exactly 13.8 births. Consequently, we will assume there will be either 13 or 14 births. Likewise we will assume there will be either 8 or 9 deaths. If a normal year has 14 births and 8 deaths, then there will be 6 more people during the year. Since humans take many years to reach childbearing age, it would be a while before the increased population would result in an increase in the number of births. On the other hand, if the death rate stayed the same as the US, then the actual number of deaths could increase gradually as the population increases. The table below shows the possible changes during the first dozen years.

Year Births Deaths Final Population

Change

0 10001 14 8 1006 62 14 8 1012 63 13 8 1018 64 14 9 1023 55 14 9 1028 56 14 9 1033 57 14 9 1038 58 13 9 1043 59 14 9 1048 5

10 14 9 1053 511 14 9 1059 612 14 9 1065 6

From this chart of the first dozen years in Steilacoom Valley, we could expect a net increase of 5 or 6 people in each year. What impact would these additional people have on the food needs and energy needs of the community?

Example 2.2: The birth rate in the US is 13.82 per 1000 and the death rate is 8.38 per 1000, consequently, the growth rate, ignoring migration, is 13.82 – 8.38 = 5.44 per 1000 or 0.544%. (If we include migration, the growth rate is 0.975%). How many more people will be in the US next year as a result of births and deaths?


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Solution 2.2: This information is useful for determining how many new human beings are being added to this country each year. Since the US population is approximately 305,000,000 then

each year. This is roughly

the equivalent of adding another Phoenix to the country every year. Phoenix is currently the fifth largest city in the country. Think of all the resources that would be needed to build a city of this size each year and all the energy needs of this many people.

It may be somewhat easier to think in terms of the total fertility rate than the birth rate. The relationship between these two is shown in the following graph. The linear equation y = 1.4196 + 7.06x can be used to estimate the birth rate, given the TFR.

Example 2.3: If the TFR of the United States dropped to the same level as Europe, 1.50, what would be the effect on the US population, ignoring migration?

Solution 2.3: Use the regression equation to find the birth rate. y = 1.4196 + 7.06xy = 1.4196 + 7.06 (1.50)y = 12.01

The growth rate would be 12.01 – 8.38 = 3.63 per 1000.

each year. A city of this

size would rank between Dallas, Texas and San Jose, California which are the 9th and 10th largest US cities.


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From these examples, we can see that the population will continue to rise which means a more rapid consumption of resources and space with a corresponding increase associated with problems related to overcrowding.

MODELING POPULATION GROWTH

The population of a country or other area can be modeled mathematically. Modeling means to find a function that can reasonably estimate the growth. Models are seldom perfect, since trying to predict future events based upon past trends is difficult, but a good model can make a reasonable estimate and thus be a useful planning tool.

Before we look at modeling populations, it is important to distinguish between two types of population growth, discrete and continuous. Species that reproduce once per year exhibit discrete growth. Such species may exhibit strange variations in population because of changes in the environment or other prey or predators. Human populations show continuous growth in that births are happening throughout the year. While it might be tempting to model population growth with an exponential equation such as was used in Chapter 1, human populations, unlike money, are subjected to the limits of the environment. Consequently, each environment has an upper population limit called the carrying capacity. While it is possible to exceed the carrying capacity temporarily, eventually the environment will bring the population back to its carrying capacity. This means that for one reason or another, either the birth rate will decline or the death rate will increase. If the birth rate is not controlled by choice, then death from starvation, disease or conflict will keep the population at its carrying capacity. This is the basic idea behind Malthusian Theory as described by Thomas Malthus (1766 – 1834).

A typical approach to modeling population growth for any population with continuous growth is with the logistic function. This function reflects the idea that when the population is low, relative to the carrying capacity, growth follows a nearly exponential model. As the population approaches the carrying capacity, growth is reduced.

Logistic equation for continuous growth:

Assumptions for the logistic formula

We assume a carrying capacity K. The population cannot exceed K There is no immigration or emigration. Increasing density depresses the rate of growth instantaneously without any time lags. The relationship between density and the rate of growth is linear.

A sample logistic curve is shown below.


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Example 2.4: A town has 400 people with a carrying capacity of 1500. If the community has a 2% growth rate, how many people will they have in 10 years?

Solution 2.4: Substitute into the logistic growth equation and simplify.

=461

Example 2.5: A city has a carrying capacity of 50,000 and a current population of 25,000. If the community has a 3% growth rate, how many people will they have in 6 years?


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Solution 2.5: Substitute into the logistic growth equation and simplify.

= 27,244

These examples are simplistic in that towns and cities are not isolated and consequently additional food and supplies can be imported to provide additional resources for people. However, if there is insufficient gasoline, then trade with the city will be far more restrictive and the carrying capacity will be of more legitimate concern.

In-Class Activity 2.1: Logistic Growth ActivityName___________________________ Effort _______/3 Attendance ___/1 Total ___/4


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Washington became a state in 1889. In 1900, the population in Washington was 518,000 (Caldbick 2010). Assume the carrying capacity of the state is 8.8 million, based on the criteria used in Steilacoom Valley (only 20% of land can be used). Assume the total fertility rate had always been the same as the US is today, 2.1 children per woman. Answer the following question to create a graph of Washington’s population between 1900 and 2700.

If T is the Total Fertility Rate then determine B, the births per 1000, by using the regression equation B = 7.06T + 1.4196.

Births per 1000 _________

Convert this to a birth rate rounded to 4 decimal places Birth Rate = _________

If the death rate is always 0.0084 (current US rate), then what is the growth rate r= _______

Use the continuous form of the logistic formula, , to make a graph of the

population through 2700. Complete the table of values then graph the population for each year in the table.

Year Population1900 02000 1002100 2002200 3002300 4002400 5002500 6002600 7002700 800

Use the logistic formula to predict Washington’s population in 2008, and then compare the results to the actual 2008 population of 6.7 million. How do you explain the difference?


Washington Population if TFR = 2.1

01,000,0002,000,0003,000,0004,000,0005,000,0006,000,0007,000,0008,000,0009,000,000

1900 2100 2300 2500 2700

Year

Popu

latio

n

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Chapter 2 Homework


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Name ___________________________ Points _______/____1a. Use the logistic formula to determine the population curve for Steilacoom Valley assuming an initial population of 1000 and a carrying capacity of 2300. Use a growth rate of 0.54% (0.0054) which is the US growth rate, without immigration. Complete the table of values for each century SV exists, then graph.

0 10001002003004005006007008009001000

1b. One thousand years is a long time for a community to exist. The United States is less than 250 years old and has seen considerable depletion of its natural resources, much of which has occurred over the last 100 years. What do you expect will happen to the US population during the next 750 years?

2. The most densely populated country is Bangladesh, with a population over 10 million and a density of about 1000 per square kilometers (CIA, 2009), If we assumed this density is the maximum for the planet, then with a total land area of about 149,000,000 km2 (Cain, 2009), this planet could have a carrying capacity of 149 billion people. If the growth rate is 0.012 and the current population is 6.7 billion, how many people could we expect on the planet in100 years if the growth can be modeled logistically?


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Chapter 3 The Algebra of Sustainability


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For a community to survive for a thousand years or more on the available land and resources will require different habits of living than exist in a culture that believes in endless growth and that can maximize the consumption of resources. The creation of Steilacoom Valley requires careful thought about what is necessary for people and what is unnecessary. Humans have created many useful technologies to which they have become accustomed. Of those that we may find highly desirable are a heated home, cooking facilities, heated water, laundry facilities and medical facilities. All of these might be considered luxuries by people in various parts of the world. Things that are not needed, although we have become accustomed to them in recent years, are telephones, electronic products, plastics and motor vehicles.

Steilacoom Valley will be designed to provide small homes with minimal energy requirements for all residents. Electrical energy will be provided by windmills. The population size will be limited to the carrying capacity. The three objectives of this chapter are to

1. determine the best shape of a home so that the home will have the maximum floor area with the minimum wall space and minimum volume of air inside to be heated

2. determine the carrying capacity3. determine the number of windmills needed

These activities will be done in class, as a group activity.

Before calculating these, we will learn the skill of dimensional analysis. This method, used frequently by chemists and engineers, is an ideal way to convert units.

Dimensional Analysis Activity


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Most numbers used in the real world have units. Units are words that clarify the number. Examples of units are gallons, meters, miles, and pounds. Sometimes two units are combined with the word “per” to give a rate. Examples of this are miles per hour and miles per gallon. The word per indicates division so that the number of miles is divided by the number of hours to get miles per hour.

This activity will focus on one skill that is used by chemists and engineers to make converting from one set of units to another set easier and more organized. The skill is called unit analysis or dimensional analysis and follows a very specific process.

Unit Conversion

We typically work with units of length, mass, volume and time or with rates such as miles per hour or cubic meters per second. Sometimes the units given in the problem are not the units we need, so it is necessary to convert from one set of units to the other. While it may be easy for some to see that the conversion of yards to feet requires multiplying the quantity in yards by 3 to get the equivalent quantity in units of feet, it is not so easy to see what must be done to convert a rate of miles per hour into one of meters per second. The skill of dimensional analysis makes even the most challenging conversions a simple process.

The key to unit conversions with dimensional analysis is unit fractions. Unit fractions are fractions with different units in the numerator and denominator but in which the value of the

numerator equals the value of the denominator. For example, the unit fractions and

have different units in the numerator and denominator (feet and yard) but 3 feet equals 1

yard. The key to using unit fractions is to recognize which units are in the numerator and which are in the denominator.

Example 1. Convert 100 yards to feet. Example 2. Convert 300 feet to yards

In both examples, the original value was written followed by a unit fraction. The original value is a numerator term (with a denominator of 1). The unit fraction was written in such a way that the units in the denominator were the same as the units of the original number, thus allowing the units to cancel. The original number is then multiplied by all numbers in the numerator and divided by all numbers in the denominator. Unit equivalencies are provided in the next table.


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Unit EquivalenciesUSCS (US Customary System) USCS – Metric Metric or SI

Length12 inches (in) = 1 foot (ft)3 feet (ft) = 1 yard (yd)1760 yards (yd) = 1 mile (mi)5280 feet (ft) = 1 mile (mi)

2.54 centimeters (cm) = 1 inch (in)1 kilometer (km) = 0.62 miles (mi)

1000 millimeters (mm) = 1 meter (m)1000 meters (m) = 1 kilometer (km)100 centimeters (cm) = 1 meter (m)

Area1 square mile (mi2)= 640 acre1 acre = 43,560 square feet (ft2)

1 hectare = 2.471 acre1 square mile (mi2) = 2.59 square kilometers (km2)

1 square kilometer (km2) = 100 hectare1 hectare = 10,000 square meters (m2)

Volume8 ounces (oz) = 1 cup (c)2 cups (c) = 1 pint (pt)2 pints (pt) = 1 quart (qt)4 quarts (qt) = 1 gallon (gal)1 cubic foot (ft3)=7.481 gallons (gal)

1 quart (qt) = 0.946 liters (L) 1000 milliliters (ml) = 1 liter (L)1000 liters (L) = 1 cubic meter (m3)

Mass16 ounces (oz) = 1 pounds (lb)2000 pounds (lb) = 1 ton

2.20 pounds (lb) = 1 kilogram (kg)1 pound (lb) = 453.6 grams (g)

1000 milligrams (mg) – 1 gram (g)1000 grams (g) – 1 kilogram (kg)1000 kilograms = 1 metric ton

Energy and Work1000 Watts = 1 kilowatt1000 calories (cal) = 1 kilocalorie (kcal) = 1 Calorie (Cal)1 kilowatt hour (kWh)= 3412 British Thermal Units (BTU)

1 calorie (cal) = 4.187 Joules (J)1 Joule (J) = 1 Watt-Second (W·S)

1 kilojoule (kJ)= 1000 joules (J)1 megajoule (mJ) = 1,000,000 joules(J)

Time60 Seconds (s) = 1 minute (min)60 minutes (min) = 1 hour (h)24 hours (h) = 1 day (d)365 days (d) = 1 year (y)


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Dimensional AnalysisName _____________________________ Points _____/16 Attendance ___/2 Total ___/18

Note: Use the exact equivalencies from the table and be precise in showing the process.

In problems 1, 2 and 3, the entire dimensional analysis problem has been provided; you only need to calculate the answer. Multiply by numbers in the numerator and divide by numbers in the denominator to obtain the converted value. Ignore the ones.

(1) 1. Lengths: In a normal home, the ceilings of a room are 8 feet above the floor. What is this distance in meters?

hint (8·12·2.54/100)

(1) 2. Area: A home contains 2000 square feet. How many square meters is the house? Remember to square the unit fractions.

hint(2000·122·2.542/1002)

(1) 3. Mixed: A person consumes approximately 2000 kilocalories per day. How many kilocalories are required by a community of 500 people for a year?

In problems 4 and 5, use the Unit Equivalencies table to put in the numbers missing from the unit fraction, then cancel units that are the same in the numerator and denominator and multiply or divide the numbers, as appropriate.

(2) 4. Energy: Household energy consumption is calculated by multiplying the amount of energy needed times the number of hours that it is used. Your electric bill is calculated based on the number of kilowatt hours (kWh) that you use. The energy requirements of most appliances are measured in watts, while the time they are used is often measured in minutes, thus it is necessary to convert from watt minutes to kilowatt hours. If you know the cost of energy, you can determine how much it costs to operate an appliance.

A 1250 watt microwave oven uses 1250 watts of energy. If it is turned on for 24 minutes during the course of a day, how much energy was used in units of kilowatt hours?


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(2) 5. Energy Costs: If a clothes dryer uses 4.5 kilowatts of energy and the dryer is operated for 1 hour and 10 minutes how much does it cost to dry the clothes if the cost of energy is $0.06 per kilowatt hour.

In problems 6 to 8, do the entire problem yourself using dimensional analysis. In all cases, show the appropriate dimensional analysis procedure as demonstrated above and then complete the multiplication.

(3) 6. Volume: A person is supposed to drink 64 ounces of water a day. How many liters is this?

(3) 7. Volume: A home contains 500 cubic meters of space. What is the volume in cubic feet? Remember to cube the unit fractions.

(3) 8. Mixed: If a piece of land used for farming can produce 4000 kilocalories of energy per day per acre, then how many kilojoules of energy does it produce per day per hectare?


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Determine the shape of homes in Steilacoom Valley

Name________________________Points _____/16 Attendance ___/4 Total ___/20

Homes in the United States range in size from under 100 m2 to over 400 m2. Given that there was a time when people lived in much smaller houses, one may wonder what the smallest sized home is that can be comfortable for a family.

(1) 1. Use dimensional analysis to determine the number of square feet in a home that has an area of 100 square meters. The equivalencies you need are 100 cm = 1 m, 2.54 cm = 1in, 12 in = 1 ft. Be careful, because of the squared units in this problem.

Assumptions for Steilacoom Valley homes: 1. Families will contain 3 or fewer people (one child at most because we are trying to limit the population).2. In a sustainable community, some items will be shared, so each house doesn’t need one. Also, the amount of “stuff” a person has can be minimized.3. Every family wants their own house, but to use less land, houses will be close to each other. 4. A home that contains the most area inside with the least amount of outer wall space is the ideal for sustainability because less material is used to build the walls and less heat is lost through the walls if there is less wall space.

Shape: What is the best shape for a home? Let’s experiment with a home that has an area of 36 square meters. The formulas we will use are A=LW, P = 2L + 2W, A = πr2, C = 2πr. Determine the perimeter for each of the following shapes:

(1) Rectangle: 1 x 36 Perimeter = (1) Rectangle: 2 x 18 Perimeter = (1) Rectangle: 3 x 12 Perimeter = (1) Rectangle: 4 x 9 Perimeter = (1) Square: 6 x 6 Perimeter =

(1) Circle: r=3.385 Circumference =

(1) 2. What do you conclude is the best shape of a house for maximizing the area while minimizing the distance around (perimeter or circumference)?

Circle 1: Rectangle Square Circle

Now consider that a house is not two dimensional (length and width) but it is three dimensional (length, width and height). Therefore, the walls and roof, all of which require material to build and all of which are sources of escaping energy, must be considered. Furthermore, the volume of air inside must also be considered as larger air volumes require greater amounts of heat. Since a square was the best rectangular shaped area, lets compare a


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square home with 8 foot high walls and a flat roof to a round home built like a dome. This is like half a sphere.

(1) 3. Convert the height of 8 feet to meters.

(1) 4. What is the volume of a 6 meter x 6 meter home that has an 8 foot ceiling?

(1) 5. What is the total area of the outer walls and roof?

If a dome (half sphere) is used with the radius of 3.385 m, then the volume of the dome can be

calculated using the formula . The area of the outer walls can be computed using

(1) 6. What is the volume of the dome?

(1) 7. What is the area of the outer walls?

8. Given that both the square and dome home have the exact same floor area, answer the following questions to determine the better design.

(1) 8a. Which has less air to heat inside? Square Dome

(1) 8b. Which has less wall area through which heat is lost? Square Dome

(1) 8c. Which is more sustainable? Square Dome


This dome, which was designed by Cloud Hidden Designs, LLC, was the winner of the Domes for the World Design Challenge in 2008. Design parameters required that the diameter of the single family homes must be less than 40 feet. The cost must be less than $2,500. The objective was to provide affordable housing in areas of the world that suffer from poverty and natural disasters that destroy the local homes (Kaslik, 2008, Domes of the World Foundation, 2009).

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Determining the Carrying Capacity of Steilacoom ValleyPart 1. Finding a function for the amount of land for homes per resident

Name____________________________Points _____/6 Attendance ___/2 Total ___/8

Show work for all problems.Public buildings

To be a vibrant community, some public facilities are necessary. These might include a library, community center, activity room/theater, laboratory and medical facility.

It would be very difficult to design public buildings as we don’t have enough information, so we will just use an estimate of 20,000 m2, which is the size of the Tacoma Dome. This should provide enough space for all public activities.

We will also estimate that workshops such as the butcher shop, bakery, furniture and cabinetry shops, etc will occupy a facility of about 10,000 m2.

Carrying Capacity

The carrying capacity of Steilacoom Valley will be determined based on the following assumptions.

1. Only 20% of all the land in Steilacoom Valley will be developed. The remaining land may be used for hiking, snowshoeing or similar activities but in general it will be the amount of land that will not be developed in anyway, ever. It will be the land that residents “allow” nature to have.2. Food will be grown to meet the annual needs of the community, but not for export or long term storage.3. Land will be needed for housing, public activities, and shops4. A safety factor of 50% will be included in the amount of land needed to account for space between buildings and other additional area.5. There will not be any motor vehicles or roads.

The total land area needed for SV residents is given by:

Land =1.5 (Housing + Public Building + Shops + Farmland)

This can be expressed as the combination of functions for which H(R) is a function for the land needed for housing and F(R) is a function for the amount of farmland needed based on the number of residents.

L(R) = 1.5 (H(R) + P + S + F(R))


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Determine the land area needed for the homes, as a function of the number of residents.

Assumption 1: Each home will house an average of 2 people.Assumption 2: All homes have an area of 50 m2.

(1) 1. Determine the radius of a round home with an area of 50 m2. Round your answer to the nearest whole number.

(1) 2. If we plan to build homes close to each other by placing them inside a square property with the side length equal to the diameter of the house plus 2 meters, so that houses are about 2 meters apart, then how much land would each home require?

(1) 3a. If there are 1000 residents, how much area will be required for the homes based on our assumption of an average of 2 people per house?

(1) 3b. If there are 1200 residents, how much area will be required for the homes based on our assumption of an average of 2 people per house?

(2) 3c. Generalize this by writing the function H(R) to show how much area will be required for the homes as a function of the number of residents for any number R. H(R) should have units of square meters. Simplify completely.

Record your answer to 3c on the top of the Part 2 before turning in this activity.

If you complete this page during class, begin Part 2 of this Carrying Capacity Activity.

Determining the Carrying Capacity of Steilacoom ValleyPart 2. Finding a function for the amount of farmland and finding the carrying capacity.


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Name____________________________Points _____/15 Attendance ___/4 Total ___/19 Show All Work

From Part 1: What is the function H(R) =

Determine the amount of farmland needed for Food Production

Assumption 1: The average person in Steilacoom Valley will consume 2,500 kilocalories per day.Assumption 2: Grain and vegetables will be grown on a piece of land only once every four years. During the three years it isn’t being used, it will be allowed to grow over (fallow) and can be used by grazing animals such as bison, sheep, lamas, goats, and poultry. These animals will be used for meat and wool, milk, eggs etc. Allowing the animals on this unused (and rotated) farmland will result in natural fertilization. Most commercial fertilizer is produced from natural gas, which we are assuming is no longer available.

Assumption 3: An estimated 6000 kilocalories can be produced per day per acre

when growing grains and vegetables. The estimate for meat is about 1200 kilocalories per day

per acre, .

4. To determine the total amount of farmland needed per person solve the two simultaneous equations.

Equation 1: Total Calories per person per day = Vegetable/grain Calories + Meat Calories2500 = 6000V + 1200MwhereV = number of acres for Vegetables/grainM = number of acres for Meat

Equation 2: M = 3V since the number of acres for meat = 3 times the amount of land for vegetables/grains.

(2) 4a. How many acres are needed per person for vegetables and grain?

(1) 4b. The total number of acres needed per person is given by N = 4V.

(2) 4c. Convert the number of acres per person to square meters per person. Round to the nearest whole number.

(2) 4c. Generalize this by writing a function F(R) for which the amount of farmland needed per person is a function of the number of residents. The units should be square meters.


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(2) 5a. Simplify our land requirement function L(R) = 1.5 [H(R) + P + S + F(R)] by using the generalized results from 3 and 4 to replace H(R) and F(R). Replace P and S with their values. Combine all like terms and distribute the 1.5. Write the most simplified form of L(R).

The units for L(R) are square meters. This function represents the land requirement for each resident, under the assumptions that have been made.

(2) 5b. Since land area is usually expressed in hectares, then rewrite the function by converting the numbers to hectares. 1 hectare = 10,000 square meters

(2) 6. The total land area of Steilacoom Valley is 80 square kilometers. Only 20% of the land will be developed for human use. What is the largest amount of land, in hectares, that could be developed by the settlers? 100 hectares = 1 square kilometer.

(2) 7. Use the Land function for the amount of land you found in problem 6 to determine the carrying capacity, by solving for R.

The Algebra of Sustainability

Energy


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Name____________________________Points _____/15 Attendance ___/4 Total ___/19

At the most basic level, survival of all living organisms is dependent upon a regular influx of energy. Most living organisms get this energy from the food they eat. It is measured in calories. Of all the species, only one has been able to create enhanced living conditions by using the earth’s storehouse of energy. This storehouse contains petroleum, coal, natural gas and uranium. By using the energy stored in these resources, humans have been able to create a world where many people can do more than simply survive. This storehouse of energy contained only a few hundred year’s supply, given the size of the world population. It allows us to refrigerate and cook food, heat our homes and water, wash clothes, use machinery and electronic products, etc. The entire motivation for the Steilacoom Valley project is to relearn how to live in a world without this stored energy, which at some point, will be insufficient. Energy, in many ways, is the key to life.

Determine the energy requirements for Steilacoom Valley.

Power, which is the output of a generator, is measured in kilowatts, energy is measured in units of kilowatt·hours (energy equals power multiplied by time). All the energy used in Steilacoom Valley will be generated using windmills.

Home Energy UseWe will make certain assumptions when determining energy requirements.

Homes are small and well insulated so they only contain a small heating element. Food is prepared in communal eating areas, not individual kitchens, but they do contain

individual small cooking appliances (burner, toaster oven). Laundry is washed in designated areas so everyone does not need a washer/dryer

Estimated daily consumption6 light bulbs 13 watts per bulb, 4 hours1 tankless water heater 10 kWh/daySpace Heater 1500 watts, 1 hours Cooking appliances 1000 watts, 0.5 hours

1. Determine the daily energy use per house in kWh. (Show work, use dimensional analysis.)

(1) Light bulbs

Water heater 10 kWh

(1) Space Heater

(1) Cooking appliances

(1) Total(2) 2. Determine the community’s daily home energy use, assuming 500 houses.


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(2) 3. If the public buildings use 10,800 kWh per day (this is approximately the amount used by the Pierce College Puyallup Campus), then how much energy is used by Steilacoom Valley?

Assume all the electrical energy will be produced by windmills. Also, assume the average wind speed is 18 mph. The turbines will produce a maximum power of 1000 KW of energy with a 54 meter blade span (Layton, 2006).

(2) 4. Use the Power Curve to estimate the actual turbine output for an average wind speed of 18 mph. Show this on the graph. Change the percent to a proportion then multiply times 1000 kW.

(1) 5. Multiply the turbine output times 24 hours to determine the average number of kWh produced by each windmill in a day.

(2) 6. How many of these windmills will be needed to meet the community needs? Round up.

(2) 7. Use a safety factor of 50% to determine how many windmills should be built. This will allow for shut down due to problems or maintenance.

Chapter 3 Homework


(American Wind Energy Association, 2009)

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Name __________________________________ Points ____/___

1. Determine the carrying capacity of Steilacoom Valley using the following changes of assumptions. All other assumptions in the carrying capacity activity will remain unchanged.

Change the average home size from 50 m2 to 70m2.Change the average number of people per house to 3.Change the calories per acre for grain/vegetables to 4,500 kcal/(day·acre)Change the kilocalories per acre for meat to 1000 kcal/( day·acre)Change the crop rotation to every 3 years, thus there are two fields for meat and one for grain/vegetables.

2. Determine the number of windmills needed for 2500 people if there is an average of 2.5 people in each home. All other assumptions in the energy activity will remain unchanged.


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3. Modern society has presented us with a paradox. A century ago, if you wanted to go somewhere relatively close to home, you either walked or used a horse. Consequently most people were relatively fit. Since then, the invention of the automobile has allowed us to travel


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greater distances in less time. We have become used to using it for even short distances. Walking to a destination became a strange concept for many (why are you walking, don’t you want to drive?). Of course driving, along with some of the other sedentary things we do, has led to lower levels of fitness. Our solution for that is to drive to a favorite fitness center then exercise on a treadmill or ride a stationary bike. While it might take longer to walk or bike to our destination, we might be able to save a lot of time (not to mention money and resources) by not needing to go to the gym. This problem can be used to determine which approach saves the most time. To do so, we need to start with some basic assumptions.

Assumptions:

Assume that work is a distance of 15 miles and that because of lights, the type of roads and congestion, the average speed is 30 miles per hour. Assume that the distance to the fitness center is 12 miles from your home, that your average speed is also 30 miles per hour and that you exercise for 1 hour and take an extra half hour for changing and showering. The time to work equals the time from work. The time to the fitness center equals the time from the fitness center. We will then compare this to bicycling to and from work every day and not using the fitness center at all.

Organize your thoughts:

To help organize our thinking, keep in mind the things we spend time doing. We spend time going to work, coming home from work, going to the fitness center, exercising, etc, and returning from the fitness center. We will determine the total time involved if we use a car and if we use a bike. You will need the formula d=rt (distance = rate·time).

Total Time = Time to work + time from work + time to fitness Center + time at fitness Center + time to home.

Let T = Total TimeW1 = Time to workW2 = Time from workF1 = Time to fitness CenterF = Time at fitness CenterF2 = Time from fitness Center

T = W1 + W2 + F1 + F + F2

Calculations (show work):

a. Find the time to work using a car, W1.


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b. Find the time from work using a car, W2

c. Find the time to the fitness center, F1.

d. Find the time from the fitness center, F2.

e. Find the Total time, T when using a car.

If we chose to bicycle to work, allowed 30 minutes for a shower after we got there and bicycled home, we would get our cardio workout and not have to use the fitness center at all, but how much time would be involved? Assume we can bicycle at 12 miles per hour.

f. Find the time it takes to bicycle to work W1. Include a half hour to shower at work.

g. Find the time it takes to bicycle home from work W2.

h. Find the Total time, T when using a bicycle.

i. Based on the assumptions presented in this problem, will driving or bicycling take the least time?

4. One of the hallmarks of suburbia is the grass yard. According to Adele Weder, writer for The Tyee, an independent daily online magazine for British Columbia, a yard is “a kind of feudal crest, marking the ability to own extravagantly useless land” (Weder 2008). For many, this part


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the property demands resources such as fertilizer, weed killers and water and after using these to help the grass grow it then requires the homeowner’s sweat and time for maintenance along with gasoline for mowing. As the price of gas and food climbs, home owners may reconsider the importance of a perfectly manicured lawn.

Suppose a lawn mowing service, aware that business is decreasing, decides to expand their services by providing a service in which homeowners convert their grass yard into productive land using permaculture. The Permaculture Institute defines permaculture as “… an ecological design system for sustainability in all aspects of human endeavor. It teaches us how [to] build natural homes, grow our own food, restore diminished landscapes and ecosystems, catch rainwater, build communities and much more” (The Permaculture Institute 2007).

In Washington there is minimal rainfall in July, August and September. One of the new services of the lawn mowing company is to calculate the anticipated water needs for the gardens they install. They then design a system to collect and store water that lands on the roof of the house during the rainy months. This is an alternative to letting the water run off to the streams and the Sound and reduces the demand on city water. The objective is to determine the water needs and the amount of rain that will be needed to store enough water.

Below is a diagram of the property of one of the company’s clients.

The property measures 80 x 136 feet. The round home has a radius of 18 feet and the shadow area is a trapezoid. The long side of the trapezoid is 80 feet, the short side is 36 feet and the height is 40 feet. Assumptions: All of the property except for the house and the shadow area will be planted and will need watering. The area that is watered will need to receive one inch of water, twice a week for 10 weeks.


Radius of home: 18 ft

N

Shadow Area

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Organize your thoughts: Find the area that must be watered by finding the area of the property, then subtracting

the shadow area and half the area of the home (there is an overlap of the shadow trapezoid and half the home).

Find the number of inches of water that must be applied. Use the number of inches of water and the area of the home to find the number of cubic

feet of water needed. This is the amount of water that must be stored. How much rain must fall on the roof during the rest of the year to collect enough? Divide the volume of water needed by the area of the roof. Convert your answer to

inches.

Calculations (show work)

a. Find the area of the property.

b. Find the area of the trapezoid shadow .

c. Find the area of the home.

d. Find the area of the property that will need to be watered.

e. Use dimensional analysis to find the number of feet of water that must be applied during the 10 week period.

f. Find the volume of water that must be applied during the 10 weeks.

g. Find the number of inches of rain that must fall on the house to accumulate enough water.

h. If the house is located in a place that receives an average of 35 inches of rain a year, will the owner be able to store enough water from roof runoff?

5. In the Presidential Debate that occurred on October 15, 2008, both candidates answered the following question asked by moderator Bob Schieffer. Would each of you give us a number, a


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specific number of how much you believe we can reduce our foreign oil imports during your first term? McCain’s answered “So I think we can easily, within seven, eight, ten years, if we put our minds to it, we can eliminate our dependence on the places in the world that harm our national security if we don't achieve our independence.” Obama’s answer was “I think that in ten years, we can reduce our dependence so that we no longer have to import oil from the Middle East or Venezuela. I think that's about a realistic timeframe.” (LA Times 2008).

To gain some appreciation for what this would mean, let’s modify the question slightly and determine the impact of reducing oil consumption in the US to a level in which we will not have to import any oil. That is, we will only use the oil pumped from wells in the United States. This is not an unreasonable assumption as there will come a time when foreign countries will want to conserve their oil resources for their own country to use and so won’t sell them to the US. We will use the timeframe of 10 years as stated by President Obama.

The graph below shows historical US petroleum production and consumption data. It is based on data from the Energy Information Administration website.

a. To model this problem, we will need a linear equation. In January 1990, the US produced 7.5 million barrels of oil per day. In January 2000, the US was only able to produce 5.8 million


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barrels of oil per day. Find the equation of the line through these two points then use the equation to predict the amount produced in 2019 (10 years after the Obama Presidency began).

b. In January, 2009, the US consumed 19.1 million barrels of oil a day. To be completely off of foreign oil in 10 years, we would have to reduce our consumption to the level you calculated in the prior question. Find the slope of the line connecting the point (2009,19.1) to the production amount you found for 2019.

The slope of the line you just calculated is the number of million barrels of petroleum used per day that we would have to reduce as a nation. Assume that driving would be reduced by the same percentage as gasoline.

c. What percent reduction would occur in the first year?

d. For every thousand cars on the road in 2009, how many could not be used in 2010?

e. For every hundred days you drive in 2009, approximately how many of those days could you drive in 2010?

f. What is the percent reduction that would occur after 10 years?

g. For every thousand cars on the road in 2009, how many could not be used in 2019?

h. If this reduction occurred, how many years would it be before there would be no need for traffic reports on the news?

i. If this reduction occurred, how would Washington State pay for the new Narrows Bridge, the Alaskan Way Viaduct and the 520 Bridge?

Chapter 4 Statistics


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If all of the energy needs of Steilacoom Valley are met with windmills, then it is critical that there are sufficient winds with which to spin the turbines. This typically requires a minimum of 10 mph winds. There are two questions that are of primary interest.

1. What is the average wind speed?

2. What proportion of time is the wind speed less than 10 mph?

The first question allows for an estimate of the possible power output of the windmills and consequently the energy available for use in the community. The second question determines the amount of time that energy is not being produced which can be used to estimate the storage capacity needed.

These two questions illustrate the need for a different type of math than algebra. Algebraic math is deterministic and as such assumes that the relationship defined by the equation is always true. Winds on the other hand are highly variable and because of this it isn’t possible to write an equation to answer either question directly. The only way to arrive at a definitive answer is by recording every wind speed for every instant of time at each turbine and then calculating either the average wind speed or the proportion of time the wind speeds were too slow.

Our goal in this unit is to understand variables that change. These variables are called random variables. Examples of random variables are wind speed and whether the wind speed is over 10 mph (yes or no). We use statistics to understand these variables so that we can make informed decisions.

Our interest is always in the population of these random variables. The population is the entire collection of the random variables. In the case of wind speeds, we want to understand all wind speeds that might affect the windmills.

To determine the average or proportion of the population would require a census. A census requires getting data from every unit or person in a population and is typically not possible because of time, money, the destructive nature of gathering data. Sometimes it is simply impossible to gather all the data.

If we assume we cannot do a census, then the best we can do is take a sample. A sample is a small collection of data taken from the population. A sample can only give us insight into the population from which it is drawn, but not from any other population. For example, wind measurements taken in winter do not give information about summer winds.

The concepts presented thus far suggest the purpose of statistics. Statistics are typically used when there is a problem or a question about a random variable that people want to understand in order to make a good decision. The decision often involves money or health or quality of life issues. To make the best decision, a person would like to know the details of the population, but the best a person can usually do is take a sample. The judgment about the entire population must be based on the results of the sample.


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While it is possible to determine the average or proportion of our sample data, it is highly unlikely that this average or proportion is the same as exists in the population. This is a critical point and the first indication that you will need a different thinking process when attempting to understand random variables as compared to algebraic variables with which you are more accustomed. For example, the algebraic equation 3x = 12 will give a solution of x = 4 for everyone. However, if the average of a population is 25 and everyone in a class took a sample of the same size from this population and found the average of their sample, in many cases, the averages of the samples would not be 25 and most students would not have the same average as other students.

To illustrate this concept, consider the 9 employees of a small company to be the entire population. The data of interest is their annual salaries in units of $1000. The salaries are 15, 20, 20, 25, 25, 25, 30, 30, and 35. The only way to actually know all the values in a population is by doing a census. As has already been discussed, doing a census is not typically possible. However, to help explain the concept, it is useful to know the values of a very small population. It is easy to determine that the average (mean) of this population is 25. Suppose, as is typically the case, you did not know the mean and decided to take a sample, with replacement, of size three from the population. In that case, there are 729 possible samples you could select and in this case there are 13 different possible sample means you could get. The distribution of these sample means is shown in Figure 4.1. Notice that there is only a 19% chance that the sample mean you select would exactly equal the mean of the population. The numbers above the bar show the number and percentage of times each sample mean would occur.

Likewise, if we flip a coin ten times, we will not always get exactly 5 tails. While the proportion of tails we should get in the long run is 0.5, we can see in the graph below that if we


Figure 4.1

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flip a coin ten times and the proportion of tails varies from 0 to 1 and will equal 0.5 only 25% of the time. Thus, the other times the sample proportion does not equal the population proportion.

Ultimately, our need for this branch of mathematics occurs when we have a question about a population for which we want to know the average or proportion. Since we cannot do a census, then we need to devise a way to use our sample to estimate that average or proportion. This will require the researcher to:

1. Formulate a specific question.2. Design a study or experiment3. Use random sampling and good data collection methods4. Graph the data5. Determine relevant numerical summaries of the sample. These are called statistics.6. Use these statistics to estimate the parameter without bias.

A parameter is a number such as a mean or proportion that summarizes all the data in a population. A statistic is a number such as a mean or proportion that summarizes all the data in a sample. In most populations, the statistics tend to occur randomly above and below the parameter as can be seen in Figures 4.1 and 4.2. If for some reason, all possible statistics we could get occur to one side of the parameter and not the other, there would be bias. This might occur if we only sampled wind measurements on windy days.

Before sampling, it is helpful to first identify the type of data that will be sampled. Data is either quantitative or qualitative. Quantitative data is the result of measurements or counts (quantities). Examples include wind speed, exam scores, number of traffic lights on the way to


Figure 4.2

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school, and how high you can jump. Qualitative data, also called categorical data, is not numeric. Examples include heads/tails, yes/no, male/female, and true/false.

For qualitative data, the parameter or statistic of interest is the proportion. The proportion

of the population, which is the parameter, is . The proportion of the sample, which is the

statistic, is . For quantitative data, the parameter or statistic of interest is the mean. The

mean of the population is symbolized with the Greek letter mu and is shown as . The

statistic, which is the mean of the sample, is symbolized with x-bar and is shown as .

Sample size is represented with an n, while the population size, which is generally not known, is represent with an N.

EXPERIMENTS AND STUDIES

Since gathering data usually involves considerable time and expense, it is beneficial to have a plan of action for collecting useful data. This plan consists of conducting an observational study or an experiment. In an observational study, units are observed and data is recorded. In an experiment, the researcher imposes a treatment on the unit, with the intent of determining if the treatment has a particular effect. For example, a researcher wanting to know about the water quality of a stream would do an observational study by taking water samples from the stream and chemically analyze them. A researcher wanting to test the effectiveness of a fertilizer would conduct an experiment by putting the fertilizer on some of the crops and not putting the fertilizer on the remainder of the crops. The crops that do not get fertilizer are called the control. The purpose of a control is to provide a contrast to the units that receive the treatment.

Observational studies would be of use to determine such things as wind speeds, energy consumption per house, or number of kilocalories produced per acre. Wind speed data would be collected at randomly selected times. Daily home energy use would be determined on randomly selected homes for randomly selected days. The number of kilocalories produced on randomly selected acres could be measured.

In cases where the researcher wants to see the effect of one random variable on another, it may be helpful to conduct an experiment. For example, in Steilacoom Valley, conservation of water and electricity would be important. A researcher could conduct an experiment to determine if the use of a shower timer will help reduce the length of showers. A shower timer can be used to indicate when 5 minutes has passed. A researcher might have some randomly selected residents use a shower timer and others not use one. All the people in the experiment would keep track of the length of their showers. In this example, shower timers would be the explanatory variable or factor and the length of the showers would be the response variable. The average length of a shower would be the parameter of interest. The levels are the possible


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outcomes for the explanatory variable which are to use or not use a shower timer. This can be shown in a one-way design layout table.

Factor: shower timer Level 1 use Level 2 not use Level 3Response Variable length of a showerParameter of interest average

This experiment could be enhanced by adding a second explanatory factor. This one could be about the shower flow rate and whether it has a high flow rate or a low flow rate. To show this, we can use a two-way design layout table.

Response Variable: length of shower

Parameter of Interest: average

Factor1: shower timeruse not use

Factor 2: shower flow rate

high flow rate Treatment 1 Treatment 2low flow rate Treatment 3 Treatment 4

Each treatment represents a different combination of shower timer and shower flow rate. Thus treatment 1 represents using a shower timer with a high flow rate shower while treatment 2 represents not using a shower timer but having a high flow rate shower, etc.

SAMPLING

The key to being able to use a sample for an unbiased estimate of a parameter is random sampling. We will learn 2 good sampling methods and discuss 2 questionable sampling methods. The general idea that differentiates a good from bad method is that the sampler does not make a choice, but leaves the selection up to a random process beyond his control.

One good sampling method is called simple random sampling (SRS). In its most basic form, this means pulling names from a hat. With larger populations, numbers are assigned to each unit in the population and a random process is used to determine which numbers are picked. In SRS sampling, every unit has an equal chance of being selected as does every sample of size n. The best way to pick these numbers is at random.org. An alternative is to use a graphing calculator or a table of random digits.

Because graphing calculators are not required for this course, we will focus on using a table of random digits to select our sample. An example is provided in Figure 4.3.

Figure 4.3. Table of Random Digits.


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Row Number

1 83984 22116 01657 83717 24799 00515 37723 23445 02705 261272 78425 65082 07792 43850 22134 76033 87273 13972 58089 125383 96268 62423 63347 09111 12079 58082 88984 76565 62765 359234 58037 43470 88497 98909 79230 36845 30325 82655 48666 554315 52354 04992 47754 31246 36779 27029 88187 19275 89632 216846 65936 11549 15979 92704 42288 07121 54938 08990 00190 814027 01849 40765 97487 56378 80291 40351 95246 58004 56115 531978 94368 20871 13867 61232 87091 67621 27560 81197 63987 011189 24504 75557 58840 99065 49850 55957 14117 62890 24961 54550

10 13283 33042 69362 92759 81354 76328 76438 29699 86996 65089

Think of this table as an endless string of digits between 0 and 9. The numbers are grouped only for visual convenience.

To use the table, determine the size of the population from which you will sample. Assign a number to each unit in the population, starting at 1 and continuing until all units have been numbered. Count the number of digits in the unit with the highest number. This count will be the number of adjacent digits you select.

For example, if there are 550 units in a population and you wish to select 5 of them, starting in row 7 (an arbitrary choice), then you will look at each consecutive group of 3 numbers (because 550 is a 3 digit number) and use them if they are less than or equal to 550. If you reach the end of the row before you have all the numbers that you need, continue onto the next row without skipping any numbers. Numbers with less than three digits will become three digit numbers by putting zeros in front of them (1 will be 001, 37 will be 037).

7 01849 40765 97487 56378 80291 40351 95246 58004 56115 531978 94368 20871 13867 61232 87091 67621 27560 81197 63987 01118

The five numbers that would be selected are 018, 494, 076, 487 and 029.

When there are some distinctive subgroups in a population, such that the variation between the subgroups may be more significant than the variation within each subgroup, then stratified sampling should be used. For example, wind speeds might vary seasonally so sampling should be done in each season. Other strata (subgroups) for other questions include gender, age groups, locations etc. Once the strata have been determined, sampling should be random within each stratum.

Two bad sampling techniques are voluntary and convenience sampling. Voluntary samples give the respondent a choice to participate. Examples include web surveys and texting in response to TV surveys. Convenience sampling is sampling those within easy access. This is not necessarily bad, but often is a problem because those within easy access do not necessarily reflect a good cross section of the population.

In-Class Activity 4.1: Simple Random Sampling and Stratified Sampling Name _________________________Points ____/10 Attendance ____/3 Total ____/13


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In the Algebra of Sustainability for Energy, we found that the average SV home will use about 17 kWh of energy per day. One might expect considerable variation for each household and each time of the year however. A survey will be conduction on randomly selected homes to determine the amount of energy they use on one particular winter day. There are a total of 500 homes, so assume homes are assigned numbers from 001 to 500. Use the table of random digits, beginning in row 2, to determine the number of the first 5 different homes to be selected. Sampling will be done without replacement.

1 46264 40104 62454 04874 43761 53866 83845 352962 30267 86651 85801 87969 94780 48478 57935 085263 53925 09051 27012 62551 48307 89610 22176 460564 31696 81575 10056 94587 45168 52099 31437 018775 74578 86989 44498 59168 46812 40226 02453 740836 28015 60523 73148 43919 52432 46019 57335 774617 94305 91470 80735 49061 02861 30607 42640 394458 75327 95465 02530 38920 87659 49129 06341 058499 08040 40445 76640 81188 53845 20344 77163 11998

10 86958 74577 91486 99892 86719 26497 62380 9255711 06218 27198 42539 11519 93614 13016 15202 1988112 03544 21447 22252 32837 30523 85918 67978 5512813 51194 30624 60991 71778 99934 03136 41273 5899314 56714 64038 54602 07145 90109 75447 25781 8568515 78489 62533 21052 03151 71401 15827 42642 5484116 32091 53866 12648 58644 22827 01960 20318 3603717 00752 13863 43137 76789 76428 40317 61597 4964518 82717 24107 32039 66705 31020 36765 33712 2907119 36872 67056 99165 73423 81358 50517 17966 5378720 61739 22959 64821 88193 13539 97828 85531 94792

(2) Simple Random Sample, first 5 homes selected ______, ______, ______, ______, ______

Stratified Sampling. Assume the 500 homes are actually divided into 4 separate locations, with 125 homes in each location, each location named after a direction (North, South, East, West). You would like to select 3 homes from each. Use row 4 to select from the North, row 8 from the South, row 12 from the East and row 16 from the West. Then list the data associated with each random number.

(2) North ______, ______, ______ Energy use data ______, ______, ______

(2) South ______, ______, ______ Energy use data ______, ______, ______

(2) East ______, ______, ______ Energy use data ______, ______, ______

(2) West ______, ______, ______ Energy use data ______, ______, ______Energy Data

North South East West North South East West North South East West


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1 13 18 21 16 46 24 17 19 14 91 19 17 24 242 14 10 22 23 47 17 13 20 22 92 22 16 13 123 25 13 18 12 48 22 17 23 15 93 11 19 13 174 25 14 16 17 49 16 12 24 16 94 10 12 20 235 27 15 17 10 50 17 12 15 15 95 20 13 20 116 20 16 22 19 51 26 19 10 24 96 12 18 22 147 22 19 10 14 52 14 15 17 22 97 18 11 20 128 22 18 10 22 53 29 16 10 24 98 25 11 20 159 15 13 12 14 54 29 12 22 13 99 15 12 10 22

10 18 11 23 17 55 21 12 13 16 100 14 10 16 1011 18 10 12 19 56 10 13 21 22 101 19 13 21 2112 27 19 21 14 57 28 14 13 16 102 21 10 19 1513 28 11 19 10 58 28 14 10 14 103 14 10 20 1314 15 18 14 11 59 19 11 13 19 104 18 14 23 1515 27 18 17 13 60 10 11 20 14 105 22 10 17 1016 26 10 11 17 61 13 10 21 20 106 15 18 19 1417 10 15 15 24 62 25 12 24 13 107 20 11 11 1918 17 11 19 21 63 19 16 19 11 108 22 14 14 1219 29 17 14 21 64 26 17 22 17 109 19 15 23 1920 18 19 18 17 65 28 18 17 10 110 19 17 11 1621 10 19 15 24 66 23 10 22 11 111 16 16 23 1422 23 16 14 22 67 19 16 15 12 112 17 11 18 2423 28 18 14 21 68 10 17 10 24 113 13 12 12 1324 21 11 10 23 69 25 12 11 17 114 26 16 10 1625 24 13 11 13 70 20 12 17 15 115 25 18 10 2326 25 19 20 13 71 29 12 13 10 116 26 12 21 1527 28 13 23 22 72 11 14 23 22 117 24 19 18 2428 11 10 16 19 73 21 18 19 15 118 14 13 15 2329 24 17 23 24 74 21 12 10 17 119 14 17 24 1630 26 12 18 20 75 29 19 17 14 120 11 15 18 1731 17 10 23 12 76 23 13 14 18 121 27 16 16 1332 15 18 23 21 77 25 18 23 24 122 21 18 12 2233 18 14 20 18 78 19 17 23 13 123 10 15 17 1634 17 14 22 12 79 14 17 14 20 124 18 10 14 1135 13 12 15 17 80 16 18 13 23 125 10 11 10 1036 22 10 17 16 81 12 11 21 2337 17 19 24 10 82 18 15 24 2238 21 16 19 10 83 28 16 12 2139 11 18 11 14 84 13 15 23 2140 26 10 20 22 85 18 12 18 1941 20 13 24 14 86 15 12 22 2242 14 18 23 24 87 22 19 19 1643 19 16 14 24 88 19 11 14 1744 26 17 14 17 89 22 10 12 2145 10 19 22 24 90 12 16 20 15

PROBABILITY

Since good sampling requires a random process, then it is only by chance that particular units or subjects become part of a sample. A different random selection would result in a different sample set of data. While this is desirable, it does mean that an understanding of basic


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probability will be required to understand the theory that allows a statistic to be used to estimate a parameter. There are three particular aspects of probability that are essential for understanding the theory behind inferential statistics, which is the process of using statistics to estimate parameters.

1. Simple probability – the probability of one outcome when one selection is made.2. Or – the probability of one outcome or another outcome when one selection is made.3. And – the probability of an outcome when more than one selection is made.

SIMPLE PROBABILITY

Probability is the proportion of times an outcome will occur over the long run. The emphasis on long term is very important.

We view probability as a fraction: . Assume all

possible ways are equally likely. Probability is always a numerical value between 0 and 1. This can be shown as 0 ≤ P(x) ≤ 1. The probability is 0 if the event cannot occur. The probability is 1 if the event is a sure thing – it occurs every time.

Our objective is to find probabilities of random processes. A random process (such as random sampling) is a repeatable process whose set of possible outcomes is known, but the exact outcome cannot be predicted with certainty. The set of possible outcomes is a sample space. A subset of the sample space is called an event.

Example: The sample space when flipping one coin is {head, tail}. An event is getting a tail.Example: The sample space for the gender of 3 children is GGG, GGB, GBG, BGG, BBG, BGB, GBB, and BBB. An event is having 2 girls.

Example 4.1: Find the probability of getting a head when flipping a coin one time.

Solution 4.1: Since there are two possible outcomes in the sample space {H,T} of which only one is favorable {H}, then

.

Example 4.2: Find the probability of a three child family having two girls.

Solution 4.2:. Since there are eight possible gender sequences {GGG, GGB, GBG, BGG, BBG, BGB, GBB, and BBB}, of which three have two girls {GGB, GBG, BGG}, then

.

P(A OR B)

If you are collecting data on whether students thought the economy, the environment or social justice was the most important component of sustainability, then a simple probability


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question would be to determine the probability that one randomly selected person favored the economy.

A more challenging question is to determine the probability that one randomly selected person favored the economy or the environment. In this case, one person is selected but there are two possible responses that would be considered favorable. We ask the question as P(Economy or Environment) or more generally as P(a or b). The probability is found by adding the simple probabilities of each outcome. Thus, P(a or b) = P(a) + P(b).

Example 4.3: A school has 4000 students and 1500 believe the economy is most important, 1400 believe the environment is most important and 1100 believe social justice is most important.A. Find the probability that one randomly selected student thinks the economy is most important.

Solution A: =

B. Find the probability that one randomly selected student thinks the economy or the environment is most important.

Solution B: = or find it

using the P(a or b) rule:

P(Economy or Environment) = P(Economy) + P(Environment) = .

Our use of the P(A or B) rule is limited to mutually exclusive events, which are events that cannot both happen at the same time. A particularly important application of this rule is with complements. Complements occur when there are only two possible outcomes. The probability of one of the possible outcomes is equal to one minus the probability of the other outcome. For example, what is the probability that a shopper will remember or not remember to take the reusable bags into the grocery store? This can be shown as P(remember or not remember) = P(remember) + P(not remember).

Since it must happen that the person will remember or not, then P(remember or not remember) = 1.

Consequently 1 = P(remember) + P(not remember).

Solving algebraically for P(remember) we get P(remember) = 1 – P(not remember). Solving algebraically for P(not remember) we get P(not remember) = 1 – P(remember).

The Complement Rule: If A and (called A complement) are the only possible outcomes and they are mutually exclusive then P(A) = 1- P( ) and P( ) = 1- P(A).


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Example 4.4: If there is a 40% chance that a shopper will forget to take the reusable bags into the store, what is the probability that the shopper will remember to take them in?

Solution 4.4: P(remember) = 1 – P(not remember)P(remember) = 1 – 0.4P(remember) = 0.6

P(A AND B)

Determining simple probabilities for one selection is necessary for being able to determine the probabilities of more than one selection. When there is more than one selection, we are interested in finding the probability of an outcome A on the first selection and an outcome B on the second selection etc. This is shown as P(A and B). Although sometimes this probability can be found using simple probabilities and sample spaces, more often it is useful to us the formula P(A and B) = P(A)P(B). This formula shows that the probability of outcome A on the first selection and outcome B on the second selection is equal to the product of their probabilities. We will assume that the events are independent, which means the first selection does not affect the probability of the second selection.

Example 4.5: If a coin is flipped 2 times, what is the probability of getting 2 heads?

Solution 4.5: This can be solved in two ways.

The first way is to create a sample space and then determine the probability. The sample

space is {HH, HT, TH, TT}. From this, we can see the probability of two heads is .

The second way is to use the rule for the probability of two events. In context, we can say that the probability of getting a head on the first flip and a head on the second flip is

P(H1 and H2) = P(H1)P(H2) = .

USING DATA TO ANSWER QUESTIONS


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The basic concepts for answering questions about a population have now been provided. These concepts include the design of studies and experiments, the random selection process, probability and the awareness that samples that could be selected from a population vary, implying that many different values could be obtained for the statistic, although there will always be a fixed, but unknown value for the parameter.

The processes for answering the researcher’s questions will be provided for both quantitative and qualitative (categorical) data. The processes are similar, although the theory that justifies each is different. The processes will be presented in the following sequence:1. Graphing2. Statistics3. Theory for making Inferences4. Confidence Interval

Inferences are made when sample data are used to infer something about the population. In this text, the only inferences to be made will be with confidence intervals.

GRAPHING QUANTITATIVE DATA

Data that has been collected is typically a chaotic collection of numbers or words that at first glance has no meaning to anyone. Consequently, the statistician needs to organize the data. The first way to organize the data is by graphing it. This allows the researcher to see how the data is distributed. Graphing gives a good visual impression of what the data suggests about the population from which it was drawn.

The type of graph we use depends upon the type of data. Pie charts are used for qualitative data while histograms are one of the primary means for graphing quantitative data.

When data is quantitative, generally a collection of measurements, then it can be graphed with a histogram. A histogram is a bar graph in which similar size measurements are grouped and counted. The x axis provides the lower and upper boundaries of each class (group) while the height of the bar indicates how many values are in each class.

Steps to make a histogram.1. Determine the lowest and highest values.2. Create reader-friendly class boundaries

a. pick a good starting point that either equals the lowest value or is less than the lowest value.

b. pick a class width (difference between consecutive lower boundaries) that will produce 4 to 10 classes.

c. Show the boundaries using interval notation, for example [10,15) which would indicate all numbers greater than or equal to 10, but less than 15.

3. Create a frequency distribution and count the number of values in each class.4. Make the histogram. Label the x axis with the lower boundaries. Label the y axis

with the counts. Include a graph title and axis titles on both axes.


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5. Look closely at the graph to determine what the graph suggests about the data, in relation to the question that prompted the research.

A note about reader-friendly intervals: A graph is a form of communication. If you take all the time and effort and money to conduct important research, then it should be important for you to communicate the results clearly. A starting value of 10, with a class width of 5, leads to reader-friendly numbers on the x axis such as 10, 15, 20, 25, etc. On the other hand, using a starting value of 10.2 with a class width of 4 would result in x axis numbers (10.2, 14.2, 18.2 etc) that are definitely not reader-friendly and would make the reader have to work too hard to understand the data. In general, the starting value should be a multiple of the class width.

Example 4.6: Suppose 20 wind measurements were taken in one area with the hope of understanding the distribution of wind speeds.

The Data:17.7 20.7 16.9 14.6 20.1

11 24.1 18.9 15.5 15.724.3 23.7 27.1 28.2 14.112.9 21 10.2 18.8 23.3

Solution 4.6:Determine the class boundaries

Lowest Value = 10.2Highest Value = 28.2Use 10 for the starting value and use a class width of 5.

Create the frequency distributionClasses Frequency[10,15) 5[15,20) 6[20,25) 7[25,30) 2

Draw the histogram

This graph suggests that wind speeds exceed 10 miles per hour, and are frequently between 20 and 25 miles per hour. This would make a good location for windmills.

STATISTICS FOR QUANTITATIVE DATA


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For quantitative data, we are interested in two different statistics. One is a number to represent the center of the data set and the other is a number to represent the variation in the data.

The most common measures of the center are the arithmetic mean and median. Up to this point we have been using the word average, but now we will be more formal and call it the mean.

The mean of a set of n observations of a quantitative variable is simply the sum of the observation values divided by the number of observations, n.

Sample Mean:

Population Mean:

The symbol ∑ is an upper case Sigma and means summation – to add up all the data values.

Example 4.7: Find the mean of the following three sets and show on a number line.Set A 1,2,3; Set B 1,2,6; Set C 1,2,12

Solution 4.7:

Set A 1,2,3

Set B 1,2,6

Set C 1,2,12

A A AB B BC C C

0 1 2 3 4 5 6 7 8 9 10 11 12

The means are shown with a circle. Notice that the mean is not allows a good representation of the data set.

Median – The median of a set of n observations, ordered from smallest to largest, is a value such that at least half of the observations are less than or equal to that value and at least half the observations are greater than or equal to that value.

The median is the middle value of ordered data. Use to find which value is in the middle

where n is the number of data values once they have been put in order from lowest to highest.


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Example 4.8: Find the median of 10, 8, 11, 3 and 12.

Solution 4.8: Put the numbers in order: 3,8,10,11,12. Since there are 5 numbers then =

. This means the median is the third number, which is 10.

Example 4.9: Then find the median value of 5, 25, 8, 10, 20, and 16.

Solution 4.9: Put the numbers in order: 5,8,10,16,20,25. Since there are 6 numbers then =

. This means the median is halfway between the third number, which is 10 and the

fourth number which is 16. Thus, the median is 13.

STANDARD DEVIATION

In addition to finding the center of a data set, we also need some idea of the spread of the data. This is determined by calculating the standard deviation. Standard deviation is approximately the average distance between each point and the mean. Consider the following two sets of data:

Set 1: 4,5,6,7,8

Set 2: 1,2,6,10,11

Make a frequency plot of both. What is the mean and median?

Do they look the same? Notice that set 2 values are spread out more than set 1 values. We would expect that the average distance each value in set 2 is from the mean is greater than the average distance each value in set 1 is from the mean. We calculate this using the formula for standard deviation. As with the other statistics, there is a difference in notation between the standard deviation of the population and the standard deviation of the sample. The two formulas are

Population standard deviation:

Sample standard deviation: s =

Because the population standard deviation (σ – lower case sigma) requires knowledge of μ, which would require a census, we will focus on the sample standard deviation (s).

When calculating standard deviation by hand, it is convenient to use a table. The first column in the table is for the data, the second column shows the difference between the mean


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and the data, the third column shows the square of the difference. The sum of the third column becomes the numerator for the formula.

x4 4-6 = -2 45 5-6 = -1 16 6-6 = 0 07 7-6 = 1 18 8-6 = 2 4

Total: 10

s = 1.58

Determine the standard deviation for set 2.

x1261011

Total:

In-Class Activity 4.2: QAW HistogramName___________________________ Points ______/ 6 Attendance ___/ 2 Total ______/8


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During the QAW project at the beginning of the quarter, all students calculated the weighted mean for their evaluation of critical issues that were presented graphically. Enter the scores for the class in the table below.

Make a frequency distribution and histogram of the data. Use reader - friendly classes, keeping in mind that the scoring system was meant to be consistent with the grading system used in schools (4 point scale). After making the histogram in the space below, find the mean and median.

Mean_______ Median ______


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THEORY


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We now know how to find the mean and standard deviation of sample data, but to answer any question that would be the reason for conducting research in the first place requires that we use the knowledge of the sample to infer something about the mean of the population. Our ability to do this will be based on the following concepts.1. Mathematical models can be used to represent irregular or unknown distributions2. The Central Limit Theorem allows for the use of the Normal distribution and leads to a confidence interval formula for estimating the parameter.

MATHEMATICAL MODELS

If you were asked to find the area of the following shape, how would you do it?

One approach would be to cover it with equally spaced grid lines then count the number of squares.

An alternate approach is to model the shape with a geometric shape of known properties. In this case, since the shape looks somewhat like a circle, we might model it with a circle.

With this circle, we can find the radius and then use the formula A = πr2 to estimate the area of the original drawing. We may not have a precise area for the original drawing, but it should be a close estimate.


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Extending this concept to data, a smooth curve has been placed on the graph that was presented in figure 4.1. The graph, with the curve is reproduced in Figure 4.4. The curve is called the Normal Distribution but also goes by the name Bell Curve.

Figure 4.4

Like the circle in the earlier example, this curve has some known properties. Among these properties are

1. The area under the curve is 1.2. The mean and median are both located in the middle of the distribution. Half the

curve is above the mean and half is below the mean.3. The curve can be labeled with 3 standard deviations on either side of the mean.4. Approximately 68% of the curve is located within 1 standard deviation of the mean.5. Approximately 95% of the curve is located within 2 standard deviations of the mean.6. Approximately 99.7% of the curve is located within 3 standard deviations of the mean.7. The area of a portion of the curve corresponds to the probability of selecting a value within that portion.


99.7%0 1-1 2-2 3-3

95%

68%

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SAMPLING DISTRIBUTION OF SAMPLE MEANS

When you learned to make a histogram of data, you were making a histogram of individual values of the random variable X. If, on the other hand, you took samples of size n from the population and found the sample means and made a histogram of the sample means, , that would be a sampling distribution of sample means. While it is not logical to create sampling distributions, it is necessary to visualize them to understand statistical inference.

Figure 4.5 shows the distribution of 1330 wind speeds, which are raw data. Notice that the data are not normally distributed and consequently is not well modeled by the normal curve.

Figure 4.5

If samples of size 36 are drawn from this distribution, then the means of these samples can form a sampling distribution of sample means. This distribution, consisting of 1000 different sample means drawn from the population of the original wind speed data, is approximately normally distributed. It is shown in Figure 4.6.

Figure 4.6


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CENTRAL LIMIT THEOREM

The Central Limit Theorem is one of the most important of all the statistical theorems. This theorem states that given any distribution with a mean μ and a standard deviation σ, the distribution of sample means will be normally distributed with mean μ and standard deviation

, provided the sample size is sufficiently large. What this implies is that regardless of the

shape of the distribution of the data, the distribution of sample means will be normal. In general this is the case if the sample size is greater than 30. If the sample size is less than or equal to 30, the original data must be normally distributed.

Figures 4.5 and 4.6 illustrate the Central Limit Theorem. In 4.5, the data are not normally distributed, but in 4.6 the sample means are normally distributed. By putting the histogram from 4.6 onto the same x-axis scale as in 4.5, it is evident in Figure 4.7 that the standard deviation of the sample means is much less than the standard deviation of the original data. This is shown because the curve is narrower.

Figure 4.7

CONFIDENCE INTERVALS

Since the distribution of sample means is normally distributed with mean μ and standard

deviation and 95% of a normal curve falls within two standard deviations, then we can

conclude that 95% of all possible sample means from a population will fall within 2 standard deviations of the mean of the population. Conversely, the mean of the population should then be within two standard deviations of 95% of all the possible sample means we could get. Therefore, if we start with the sample mean, which would be determined from the sample data and add and subtract 2 standard deviations, then we create an interval that has a good chance of containing


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the parameter μ. Mathematically this is shown as . There is one slight problem with

this formula. We don’t know the value of σ. Therefore, we estimate its value with s, the sample

standard deviation. The formula we use for finding the 95% confidence interval is .

Example 4.10: To foster a sense of community in Steilacoom Valley and to reduce energy demand, meals are eaten together in group dining areas. A dining area manager wants to know the average number of people who eat their breakfast at that dining area so that they can prepare enough food, without producing too much waste. A random sample of 36 days from the prior year shows the number of people at breakfast.

213 214 260 171 257 196183 163 152 218 187 219282 147 253 252 171 155138 216 239 173 181 212321 227 190 199 115 127153 125 293 124 117 267

The mean of this data is 197.5. The sample standard deviation is 53.2. The sample size is 36.

Solution 4.10:

The 95% confidence interval is

or 179.8 < μ < 215.2

Conclusion: About 95% of the sample means we could get produce a confidence interval that will contain the mean of the population. Based on our sample, we estimate that the mean number of people who come for breakfast is between 179.8 and 215.2.


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In-Class Activity 4.3: Analyzing Quantitative DataName_______________________________Effort____/3Attendance ___/1 Total ___/4

Suppose that instead of the formal educational system used in much of the world, all people were encouraged to be active learners, regardless of age. To achieve that goal, the community’s library became a major resource. Everyone, regardless of age, was encouraged to read whatever they wanted. There were no recommended books (other than by friends) and no book was considered too advanced or too simple for anyone. There were no book reports or exams. The librarians kept record of the number of books read per month by all the residents that were old enough to read. A sample of this data is provided in the table below.

1 0 26 3 723 10 5 10 259 3 8 20 1714 11 5 1 196 5 10 18 54 4 6 17 9

1. Make a frequency distribution for the number of books read per month2. Make a complete histogram for the number of books read per month3. Find the mean4. Find the median5. The sample standard deviation is 7.4. Find the 95% confidence interval.


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GRAPHS AND STATISTICS FOR QUALITATIVE (CATEGORICAL) DATA

Winds need to be over 10 mph to cause the blades of a windmill to spin. If the windmill is not spinning, electricity is not being produced. If windmills are the sole source of electrical energy, then the amount of time the turbine doesn’t spin because the winds are too slow is an issue of concern. Suppose the engineers want to locate the windmills in a place where the winds are too slow only 10% of the time. They monitored winds in one location and found that out of a sample of 200 wind measurements taken at 150 above the ground, 53 were less than 10 mph.

One graph that is used for qualitative data is a pie chart. To make a pie chart, determine the proportion of each category; multiply by 360 then estimate that number of degrees on a

circle. The statistic, which is the sample proportion is given by .

Category x n Sample Proportion

degrees

Slow winds <10mph

53 200 0.265·360 = 95.4°

Fast winds ≥10mph

147 200 0.735·360 = 264.6°

This is shown in the graph in Figure 4.8.Figure 4.8

From this graph, it does not appear that the wind measurements were taken in a good location for a windmill because the winds were too slow 27% of the time, which far exceeds the engineer’s desire. However, we must realize that this is sample data, it is not the parameter. Therefore we need to use these results to estimate the parameter which is the proportion of all wind speeds that are too slow.

THEORY


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The theory that allows for the estimation of a population proportion is different than the theory that allows for the estimation of a population mean, although both are based on the concept of sampling distributions. For proportions we use the distribution of sample proportions. The sample proportions are the proportion of successes for a particular sample size. To create these distributions we need to remember the three probability rules discussed earlier.

, P(A and B) = P(A)P(B), P( ) = 1-P(A).

Instead of presenting this in terms of coin flips that yield heads or tails, the concept will be presented more generally using the terms success and failure. These terms are not used in the traditional sense of success being good and failure being bad. Usually success is based upon the research question. If the question was “what proportion of the native salmon has sea lice as a result of a nearby salmon farm” then success would be considered the salmon with sea lice even though that is a bad thing.

Our ability to understand the characteristics of a sampling distribution require us to first use a distribution for which the proportion of successes is known. From this we will be able to see the possible sample proportions that can be obtained. We will then be able to use a sample proportion as a way of estimating an unknown population proportion. The process will be explained first using counts of successes, but will conclude by using proportion of successes.

Suppose 60% of a population can be called a success. Then 40% of that population would be a failure. If a sample of size 2 was made from this population, then the sample space could be shown as {SS,SF,FS,FF}. The probability of each would be calculated using the formula P(A and B) = P(A)P(B).

P(SS) = P(S)P(S) = (0.6)(0.6) = 0.36P(SF) = P(S)P(F) = (0.6)(0.4) = 0.24P(FS) = P(F)P(S) = (0.4)(0.6) = 0.24P(FF) = P(F)P(F) = (0.4)(0.4) = 0.16

The probability of 0 successes is 0.16, the probability of 1 success is 0.48 (0.24 + 0.24), the probability of 2 successes is 0.36. This is a small example to illustrate the point.

We can think of the number of successes in terms of proportions, where . If there

are no successes in 2 selections, then ; the probability that p=0 is 0.16. If there is one

success in 2 selections, then ; the probability that p=0.5 is 0.48. If there are two

successes in 2 selections, then ; the probability that p = 1 is 0.36.

A larger example was shown earlier in the chapter and is reproduced below (Figure 4.9). It shows all possible results of flipping a coin 10 times. In this example, tails are considered a


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success. To interpret the graph it is helpful to know that there is one occurrences for having 0 tails ( ) which can be shown as P(HHHHHHHHHH). The second bar represents 1 tail (

) which can be shown in 10 different ways, such as P(THHHHHHHHH) or P(HTHHHHHHHH) etc. The remainder of the bars represents other combinations of heads and tails. Ultimately, there are 1024 different possible ways that the heads and tails can be ordered.

CONFIDENCE INTERVAL FOR PROPORTIONS

Obviously, it could be very tedious to figure out all these possibilities. It would be nearly impossible to do so for a large sample. Fortunately, we notice that the distribution of sample proportions that are formed can be approximated with a normal distribution. The mean of the normal distribution is equal to the population proportion p. The standard deviation of the normal

distribution is . Using the same logic as we did with creating a confidence interval

for the mean, we conclude that 95% of all sample proportions are within 2 standard deviations of the population proportion. Conversely, the population proportion is within 2 standard deviations

of 95% of all the sample proportions. This can be shown with the formula .

Since we don’t know the value of p, we estimate it with the sample proportion . This results in the formula we will use for the confidence interval for a population proportion,

.

Example 4.11: Find the 95% confidence interval for the proportion of wind speeds that were too slow if 53 out of 200 wind speeds were too slow.

Solution 4.11:


Figure 4.9

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= 0.265

or 0.203 < p < 0.327. Thus we estimate the proportion of all wind speeds that are too slow is between 20.3% and 32.7%. Since a location should have less than 10% of the wind speeds that are too slow, this location is not desirable.

In-Class Activity 4.4: Analyzing Qualitative Data

Name _________________________Effort____/3 Attendance ___/1 Total ___/4


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Gallup regularly asks the question: With which one of these statements about the environment and the economy do you most agree – “protection of the environment should be given priority, even at the risk of curbing economic growth (or) economic growth should be given priority, even if the environment suffers to some extent? (Gallup 2010)

From 1985 through 2008, more people favored the environment, however the survey in 2009 showed that more favored the economy. While the actual numbers are not available from Gallup, based on the statistics they provide we can use relatively realistic numbers. Suppose that 1200 people were surveyed and had an opinion and 542 of these favored the environment.

1. Make a pie chart.2. What is the sample proportion of those who favor the environment?3. What is the 95% confidence interval for the proportion of people in the country who favor the environment?


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Chapter 4 HomeworkName_________________________________________ Points ___/___


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1. As the manager of a store that hopes to sell solar panels, you have the responsibility of comparing two different brands of panels, Apollo brand and Maui brand. (Apollo is the Greek sun god and Maui is the Polynesian sun god). Both brands are rated as 1 kilowatt systems. Your goal is to determine which of the two produces the most energy after at least one year of use. To find out, you will contact people who have the system on their home. The data you will collect is the maximum daily energy output (watts). You would like an estimate of the mean energy production by each brand.

1a. Is this an observational study or an experiment?

1b. Is the data quantitative or qualitative?

1c. How many factors are there?

1d. How many levels are there?

1e. Complete the design layout table.

Factor: Level 1 Level 2 Level 3Response VariableParameter of interest

The data for the Apollo brand is 956, 890, 1000, 988 and 966.

1f. Find the mean

1g. Find the sample standard deviation

1h. Find the median

2. One way in which people will survive peak oil is by developing transition towns. In these towns, people will work together as a community to grow and preserve food and to share tools and other resources. Transition towns involve a different psychology than our current towns in which each person is focused primarily on the needs of their family, rather than of the community. As long as gas is relatively inexpensive, most people in a town won’t want to


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consider changes, but if the price goes up and the economy crashes, more people will look for alternative means of surviving.

Suppose a community decided that any edible food that is grown in front of a house could be picked and used by any community member while any food in the back of the house would be reserved for the home owner. This would be a very strange concept in the US because private property and fear of law suits would make people scared. However, assume this community solved the law suits problem and 44 home owners started planting vegetables in their front yard. Of these, 22 were randomly selected to have a sign in the front yard inviting people to help themselves when the vegetables are ready for harvest. The remaining 22 did not have the sign. The objective is to find if the signs are necessary to encourage people to pick the vegetables. Each home owner kept track of the number of visitors they had during the summer. The goal was to determine if the signs made a difference by comparing the average number of visitors at the homes with signs to the average number of visitors at the homes without signs.

2a. If the 44 home owners were assigned a number between 01 and 44, what would be the first 5 randomly selected numbers starting in row three of the table of random digits in Figure 4.3.

2b. Is this an observational study or an experiment?

2c. Complete the design layout table using the underlined words.Factor: Level 1 Level 2 Level 3Response VariableParameter of interest

2d. If the data consists of the number of people who pick vegetables at each house, is the data qualitative or quantitative?

In one neighborhood there were 8 homes with gardens. The number of visitors who picked vegetables at these homes was 3, 12, 0, 5, 0, 8, 15, 13.

2e. What is the sample mean number of visitors?

2f. What is the median number of visitors?

2g. What is the sample standard deviation for the number of visitors?3. For those who have adapted to the high stress, achievement oriented, modern day world, life in Steilacoom Valley might seem rather slow. There would probably be more leisure time than we are used to and TV would not be available for filling the time. A psychology researcher wants to find how the residents are coping with a less stressed life. Assume there are 800 adult residents and the researcher wants to randomly select 150 residents from this group. Each


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resident is given a number between 001 and 800. The following two questions were on the survey.

1. Do you want to return to the pace of life you had before moving to Steilacoom Valley?

2. How much time did you spend yesterday in conversation with your spouse/partner?

3a. Use Table 4.3, row 8 to randomly select the numbers of the first 3 people who will be asked to take the survey.

3b. Is this an observational study or an experiment?

3c. Is the data in question 1 qualitative or quantitative?

3d. What is the symbol for the parameter of interest in question 1?

3e. What is the symbol for the statistic that can be found for question 1?

3f. Is the data in question 2 qualitative or quantitative?

3g What is the symbol for the parameter of interest in question 2?

3h. What is the symbol for the statistic that can be found for question 2?

3i. The data given in response to question 1 was 25 out of 150 people answered yes. What proportion of the people in the sample want to return to the pace of life they had before moving to SV?

The data given in response to question 2 was: 10, 130, 70, 100, and 80.

3j. What is the mean amount of time couples spent in conversation?

3k. What is the median amount of time couples spent in conversation?

3l. What is the sample standard deviation of the amount of time couples spent in conversation?

4. In Steilacoom Valley, careful records are kept of the agricultural yield. The original

expectation was that 6000 calories could be produced per day per acre, . With about

1000 acres of farmland, an estimate of the total yield can be obtained by sampling 15 acres. The results of this sampling are shown in the table below. The residents of SV have a particular interest in this because it affects their year long food supply.


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5285 6680 5940 5382 61575709 6763 5666 4525 66776551 5244 5876 5711 5563

4a. Make a frequency distribution and histogram for this sample data.

4b. Find the sample mean.

4c. If the sample standard deviation is 633.4, what is the 95% confidence interval for the mean yield?

4d. As a resident of SV, what do you think are the implications of this result?

5. The Steilacoom River has an abundant salmon run each year. While salmon are a good source of protein and omega oil, fishing sustainably is important for the resource to be available in future years. Consequently, the community strongly supports the concept of not keeping the largest salmon. Since fishing will only be done for food, not for trophies to put on the wall, leaving the biggest salmon to spawn will mean that the genes of large salmon will be passed on to future generations. If these fish were removed, then over time, the salmon would become


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smaller. Out of 136 salmon that were caught this year, 28 were returned because they were too big.

5a. Make a pie chart for this data

5b. What is the sample proportion?

5c. What is the 95% confidence interval for the proportion of all the salmon in the run that are too big?

6. The Steilacoom Valley Bakery attempts to produce just enough bread so that there is no waste. Unfortunately, that means that some days there won’t be enough bread for everyone. Of 286 randomly selected days, there were 42 days in which there was surplus bread at the end of the day.

6a. Make a pie chart for this data

6b. What is the sample proportion?

6c. What is the 95% confidence interval for the proportion of all days there are surplus bread?

7. It is estimated that each home in SV will consume an average of 17 kWh per day. A random sample of homes resulted in the following set of data.

19.0 13.0 12.8 17.1 18.721.1 17.3 20.3 11.5 17.416.0 19.0 13.5 17.8 19.5


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18.1 16.5 16.9 19.0 15.1

7a. Make a histogram for this sample data.

7b. Find the sample mean.

7c. The sample standard deviation is 2.6; what is the 95% confidence interval for the mean amount of energy?

7d. As a resident of SV, what do you think are the implications of this result with regards to the number of windmills we planned for?

Chapter 5 System Dynamics Modeling

The world in which we live is very complex. Most news items look at one issue in isolation from all other issues. For example, health care is not discussed in the same report as wealth distribution. The growth of the gross national product is not discussed in the same report as resource depletion. Likewise, worldwide hunger is not discussed with global climate change.


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This book started with the Quantitative Assessment of the World project as a way to begin the process of thinking of multiple issues at the same time. During the Consequences project you had the opportunity to experiment with system dynamics modeling on a very elementary level. This chapter will expand the concepts for modeling to help you understand the tools that are used to model large natural and manmade systems such as global climate change and resource depletion.

Through much of your math education you have been taught one concept at a time with no need to be concerned about incorporating other concepts or issues. In the chapter on the algebra of sustainability you had to draw upon a variety of algebraic skills and solve a variety of connected problems in order to arrive at a meaningful solution. This chapter expands that thought process by requiring you to find the connections between non-mathematical concepts and then show these connections mathematically. This will mean you have to think, not simply look for a process to follow.

The objective of this chapter will be to expand your knowledge of system dynamics modeling and then use a computer to model the outcome on a system. This chapter will still be rather elementary in its approach, but will expand the concepts of stocks and flows that were presented earlier. Our modeling will be done using the spreadsheet Microsoft Excel. There are other software programs specifically designed for system dynamics modeling. These include Stella, Dynamo, Powersim, and Vensim.

STOCKS, FLOWS AND CONVERTERS

In our earlier experience with system dynamics modeling, you were introduced to stocks and flows. This time we will also include converters.

Stocks are used to represent a quantity of something of importance to the system. They are a quantity that will change slowly. A stock will remain at its current value if all flows cease. We can define stocks for some of the chapters in this book.

Table 5.1Chapter Stock Stock Units System Time1 - Finances Money in an account Dollars Year2. Population Number of residents in

Steilacoom ValleyPeople Year

3. Food Amount of food Kilocalories Year

To understand why these are stocks, see if they change slowly, would not change if the flows stopped and have units that represent a quantity. We also need to consider the time period for the changes that occur in a system. For all three of these examples, we were concerned with annual changes in our bank account, the population or food.


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Money in an account changes slowly when interest is added. If no interest, deposits or withdrawals are made, the account balance does not change at all. Consequently, the account balance is a stock. Dollars are units that represent the quantity of money.

The number of people in Steilacoom Valley is a stock because the population changes slowly with each birth, death or migration. If there are no births, deaths or migration, the population would not change at all. People are the units that represent a quantity of residents.

The amount of food is also a stock because the amount of food changes slowly with growing, harvesting, hunting, fishing and preserving along with consumption and spoilage. If all of these ceased, the food stocks would remain unchanged. Food could be measured in a variety of units, but in that chapter, they were measured in kilocalories.

The symbol for stocks is a rectangular box.

Flows are things that can make stocks change. To determine the flows requires an understanding of the system being modeled. In Table 5.2, the flows have been added to our examples from the book

Table 5.2Chapter Stock Stock Units Flows Flow Units1 - Finances Money in an account Dollars Deposits, interest,

withdrawalsDollars per year

2. Population Number of residents in Steilacoom Valley

People People Added, People Leaving

People per year

3. Food Amount of food Kilocalories Food Produced,Food consumed

Kilocalories per year

The units for flow are based on the stock units and the time interval for the system. They

are shown as a rate, .

The symbol for flow is a circle with a bar that connects to the double line that indicates the flow of materials.

Converters are values that help explain the flow. For interest added to an account, a converter would be the interest rate. For people added to a population, the converter would be birth rate or immigration rate. These are shown in Table 5.3.

Table 5.3Chapter Stock Stock Units Flows Flow Units Converters1 - Finances

Money in an account

Dollars Deposits, interest,

Dollars per year

Interest rateCompounding


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withdrawals Period2. Population

Number of residents in Steilacoom Valley

People People AddedPeople Leaving

People per year

Birth rateDeath rateimmigration rateemigration rate

3. Food Amount of food

Kilocalories Food Produced,Food consumed

Kilocalories per year

Fruits, nuts, vegetables and grainsRaised animalsWild game and fishEating rateSpoilage rate

The symbol for converters is a circle.

Each symbol in the model should be labeled for clarity.

Clouds are used to show stocks that are not explained. They can serve as a starting point or ending point for a model. Connectors are arrows that indicate the relationship between converters, flows and stocks.

The model for finances

The cloud represents an undefined source of money that goes into the account. Interest is added to the account. The amount of interest is determined by both the interest rate and the amount of money in the account.

Model for the population of Steilacoom Valley.


Steilacoom Valley Population

Account Balance

Interest Added

Interest RateCompounding Period

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Model for Food

CAUSAL LOOP DIAGRAMS

After making the stock and flow diagrams, the next step is to make the causal loop diagrams. This allows for a more complete understanding of the system by detailing the various feedback loops that exist and determining if they add to the stock or deplete the stock.

Causal loop diagrams should show loops. Each arrow in the loop is labeled with a plus or minus sign to indicate its polarity. A plus sign indicates that the causal effect is positive. This means that an increase in one variable causes an increase in the other. Likewise, a decrease in the one variable causes a decrease in the other. A negative sign indicates the causal effect is negative. This means that an increase in one variable causes a decrease in the other. Likewise, a decrease in the one variable causes an increase in the other.

To build causal loop diagrams, start with the stocks, then add the flows and determine if the effect of each relationship is positive or negative.

The model for finances


People Added

Birth Rate

Immigrations Rate

People Leaving

Death Rate Emigrations Rate

Food

Food Produced

Fruits, nuts, vegetables and grains

Food Consumed

Eating Rate Spoilage RateWild game and fish Raised

(chicken, etc)

Account Balance

Interest Added

Interest RateCompounding Period

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All the arrows have a positive polarity. As the interest rate increases, so does the amount of interest that is added to the account. More frequent compounding leads to more interest added. As the amount of interest increases, so does the account balance. As the account balance increases, so does the amount of interest (compounding). Because all polarities are positive, this is a positive feedback loop so it is labeled with a plus sign in the middle of the loop.

More specifically, a loop is a positive feedback loop if there are either no negative polarity arrows or there are an even number of negative polarity arrows. If there are an odd number of negative polarity arrows, the loop will be a negative feedback loop.

Model for the population of Steilacoom Valley.


Account Balance

Interest Added

Interest Rate+

+

+ +Compounding Period

+

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An explanation of each causal arrow, assuming every other value remains unchanged.Higher birth rates lead to more people added Higher death rates lead to more people leavingHigher immigration rates leads to more people added.

Higher emigration rates lead to more people leaving

More people added leads to higher SV Population

More people leaving leads to smaller SV population

Higher population leads to more people added (more births)

A higher SV population leads to more people leaving (more deaths).

This is a positive feedback loop because there are no negative polarity arrows

This is a negative feedback loop because there is one arrow with negative polarity.

COMPUTER MODELING

The final step of this process will be to model the system. This modeling will be done using Microsoft Excel. The first example will be the finance model. We will design the spreadsheet to calculate the value of the stock (account balance) for the first 30 years. These values will be computed for three interest rates, 3%, 5% and 7%. Ultimately, we will graph the account balances for all three interest rates on the same graph so we can compare the results.



People Added

Birth Rate

Immigrations Rate

People Leaving

Death Rate Emigrations Rate


People Added

Birth Rates

Immigration Rate

+

++

+

People Leaving+-

Death Rate

Emigration Rate

-++

+

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Sample Spreadsheet Design for Comparing the Account Balance of an Investment

A B C D E F G H I J K L M1 Constant Flows Time Stock2

Interest Rate 0.03 Interest Added YearsAccount Balance

Comparison of balances with different interest rates

3 Compounding frequency 12 0 1000 Years

Balance (r=0.03)

Balance (r=0.05)

Balance (r=0.07)

4 =G3*((1+$B$2/$B$3)^$B$3-1) 1 =G3+D45 =G4*((1+$B$2/$B$3)^$B$3-1) 2 =G4+D5

After typing in the formulas, highlight D4:G5 then autofill until you reach year 30, which will occur in row 33. Copy and paste the year values in column J then copy and paste special – values the account balances in column K. Change the interest rate to 0.05 then copy and paste special – values the account balances in column L. Repeat with a 7% interest rate and paste into column M. Make an xy scatter graph of columns J-M. Label completely.


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Modeling the population in Steilacoom Valley is similar to, yet somewhat different than the financial model. The financial model only includes the addition of interest. For the population model, it is necessary to consider births and immigration which add people to the Valley and deaths and emigration which remove people from the valley. For this modeling, we will assume the immigrants will only be accepted as replacement for emigrants, consequently the two values will be equal. We will also assume that death rates remain constant.

The primary complication with this model is birth rates. Birth rates tend to decrease as the carrying capacity is reached. Remember that we found the carrying capacity of Steilacoom Valley to be 2494? For our model, we will round this to 2500. We need a way to model the changing birth rates. This will be done by multiplying the birth rate by a value that varies from 1 to 0 and decreases as the population increases. The usual approach would be to use a linear model for the multiplier. We will compare that to an exponential decay model. To do so, we will generate the equation of the line that connects the points (0,1) to the point (2500,0.001). As ordered pairs, the x coordinate represents the population and the y coordinate represents the multiplier.

Notice that both these points have some flaws that we will ignore, because they will still provide a good estimate. Assume the birth rate if 5%. The first point says that if there are no people in Steilacoom Valley, the birth rate will still be 5%. However, if there are no people in Steilacoom Valley, there obviously cannot be any birthrate. The other point (2500,0.001) would ideally be written as (2500,0) because the birth rate would decline to zero when the carrying capacity has been reached. However, since an exponential function cannot have a value of 0, it is necessary to modify that value to 0.001. These issues do not alter the fact that these models can still help us understand the system.

Figure 5.1

The equation for the linear model is Multiplier = 1 – 0.0004x. The equation for the exponential model is Multiplier = 1e-0.0028x.


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Sample Spreadsheet Design for Comparing the Effect of Birth Rate Multipliers on Steilacoom Valley Population

After entering the formulas in the table, highlight C4:G5 then autofill until you have 200 years of computations. Copy the Time and Population values from columns F and G and paste them in another column using paste special – values. Then change the formula for the birth rate multiplier to the exponential model typing the formula: =exp(-0.0028*G3) into cell C4 and then autofilling.

Graph both the linear model and the exponential model on the same graph, with years as the x axis.

Does the linear model result in a graph that most resembles exponential growth, logistic growth, neither of these?

Does the exponential model result in a graph that most resembles exponential growth, logistic growth, neither of these?

Using the linear model, one might expect the population to reach the carrying capacity, but it does not. The system reached a point of dynamic equilibrium at about 80% of the carrying capacity because births and deaths were equal.


A B C D E F G1 Constant Flows Time Stock2 Birth Rate 0.05 Birth Rate Multiplier People Added People Leaving Years Population3 Death Rate 0.01 0 10004 Immigrate Rate 0.01 =1-0.0004*(G3) =(C4*$B$2+$B$4)*G3 =($B$3+$B$5)*G3 1 =G3+D4-E45 Emigration Rate 0.01 =1-0.0004*(G4) =(C5*$B$2+$B$4)*G4 =($B$3+$B$5)*G4 2 =G4+D5-E5

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In-Class Activity 5.1 Systems Dynamics Modeling for Weight

Name_______________________________ _____/9 Attendance ___/3 Total ___/12

Before modeling a system, comprehensive research is necessary to understand variables that affect the system. For this activity, the topic of body weight will be used because many people already have a basic understanding of issues related to body weight. In this activity, you will be given the stock, flows and converters which you will use to make a stock and flow diagram, a causal loop diagram and an Excel spreadsheet that will allow you to compare the effect on body weight if someone lives in an area where they drive everywhere compared to if they can walk everywhere.

The stock for this system is body weight, measured in pounds. The time period will be one week. The flows for the system will be energy consumed (eating) and energy burned. Energy is normally measured in calories, but you will convert it to pounds so that flows can be measured in pounds per week. The two converters will be resting energy expenditure and activity.

For this system, you will need to know that 3,500 Calories equates to 1 pound of fat. You will also need to find the resting energy expenditure (REE) which is based on the Mifflin Formula (Mifflin 1990).

Women: BMR = (10 x weight in pounds) + (6.25 x height in inches) – (5 x age in years) - 161Men: BMR = (10 x weight in pounds) + (6.25 x height in inches ) – (5 x age in years) + 5

(3) 1. Make a stock and flow model for this system.

(3) 2. Make a Causal Loop Diagram for this system.


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Answer the follow questions with a + or - to help determine the polarity for each causal arrow.1. How does eating more affect body weight?2. How does body weight affect eating more?3. How does burning more calories affect body weight?4. How does body weight affect burning calories?5. How does burning calories affect eating more?

(3) 3. Make an Excel spreadsheet for this system that spans 52 weeks.Assumptions: Individual is a 6 ft tall, 200 pound male who is 25 years old.Compare the weight on a graph for the following two scenarios

1. The individual drives to all destinations and does no other exercise. In this case, the individual will consume 2500 calories per day or 17,500 calories per week.

2. The individual walks to all destinations averaging 10 miles of walking a week. Assume 100 Calories are burned per mile. In this case, the individual will consume 3100 calories per day because the walking makes him hungrier.


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Appendix


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Caldbick, J. March 03, 2010. HistoryLink.org <http://www.historylink.org/index.cfm?DisplayPage=output.cfm&file_id=9332>. Accessed 2010, April 1.

[CIA] The World Factbook 2009. Washington, DC: Central Intelligence Agency, 2009.https://www.cia.gov/library/publications/the-world-factbook/index.html Accessed 2009, September.

[CIA] The World Factbook 2010. Washington, DC: Central Intelligence Agency, 2010.https://www.cia.gov/library/publications/the-world-factbook/index.html Accessed 2010, March 31.

Darwin, CG. 1953. The Next Million Years. Garden City, NY. Doubleday. 210 p

Davis, W. 2010. The Long Now Foundation Podcast: The Wayfinders: Why Ancient Wisdom Matters in the Modern World (podcast-2010-01-13-davis.mp3). The Long Now Foundation. http://www.longnow.org/projects/seminars/SALT.xml. Accessed 2010 March 31.

Domes for the World Foundation. 2009. http://www.dftw.org/stories/design-challenge. Accessed 2009, July 30.

Duncan, RC. Ph.D. 2000. The Peak Of World Oil Production And The Road To The Olduvai Gorge, Dieoff.org. <http://dieoff.org/page224.htm>. Accessed 2010 March 31.

Gallup 2009. Environment. 4 Sep. 2009. <http://www.gallup.com/poll/1615/Environment.aspx>. Accessed 2009, Sept 15.

Jordan, C. 2009. Chris Jordan Photographic Arts <http://www.chrisjordan.com/>. Accessed 2010, Apr 12.

Kaslik, J. 2008, Dec 10. Design Contest Results. [Personal Email]. Accessed 2008, Dec 10.

LA Times October 2008. Complete Final Debate Transcript: John McCain and Barack Obama. <http://latimesblogs.latimes.com/washington/2008/10/debate-transcri.html>. Accessed 2010, April 2.


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Layton, J. "How Wind Power Works." 09 August 2006. HowStuffWorks.com. <http://science.howstuffworks.com/wind-power.htm> 31 March 2010.

Hoyle, F (1964). Of Men and Galaxies. Seattle. University of Washington Press. 73 p.

Mifflin MD, Jeor ST, Hill LA, Scott BJ Daugherty SA and Koh YO. Feb 1990.A new predictive equation for resting energy expenditure in healthy individualsAm. J. Clinical Nutrition; 51: 241 - 247.< http://www.ajcn.org/>. Accessed 2010, April 2.

The Permaculture Institute. 2007. Permaculture Defined. <http://www.permaculture.org/nm/index.php/site/classroom/>. Accessed 2010, April 2.

Weder, A. 2008. The Tyee. A City's Shapes to Come? Rethinking the micro-house at the Movers and Shapers show. April 23, 2008. <http://thetyee.ca/Views/2008/04/23/CityShaping/>. Accessed 2010, April 2.


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Additional References

Aliaga M, Gunderson B. 2006. Interactive Statistics 3rd Edition. Upper Saddle River, New Jersey: Pearson Prentice Hall. 1014 p.

Comap. 2000. For All Practical Purposes, 5th Edition. New York: W.H. Freeman and Company 784 p.

Ford, A, 1999. Modeling the Environment An Introduction to System Dynamics Models of Environmental Systems. Washington DC, Island Press. 401 p.

Herbst AS, Schiller DJ. 1988. Avogadro’s Numbers. Butte, Montana: Indelible Ink. 96 p.

Lial ML., Hungerford T, and Holcomb JP Jr. 2006. Mathematics with applications in the management, natural, and social sciences. Boston: Addison Wesley.

Miller, GT Jr. 1997. Environmental Science. Sixth Edition. Belmont, CA: Wadsworth Publishing Company. 517 p.

MyCashNow 2009. MyCashNow Homepage. <http://www.mycashnow.com/>. Accessed 2009, July 20.

Quinn TJ., Richard DB. 1999. Quantitative fish dynamics. New York: Oxford UP. 542 p.

Raven PH., Berg LR , Johnson GB. 1998. Environment. Second Edition. Fort Worth: Saunders College Pub. 579 p.

Triola, M. 2001. Elementary Statistics, Eighth Edition. Boston: Addison-Wesley Longman, Inc. 855 p.


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