Problem Solving and Quantitative Skills in AP Biology

Problem Solving and Quantitative Skills in AP BiologyAka:We need to learn statistics.

Why?We need to use the tools, data collection techniques, models, and theories of science toproperly analyze our results. We will learn to be real SCIENTISTS!!!!Up until now, there is strong possibility that you have you used the word hypothesis in only one way. Now we have a null hypothesis that we either reject or fail to rejectAnd an alternate hypothesis that wereject or fail to reject.

Example:My null hypothesis at this point:

There is no difference in the statistics skill level between members of our class that reviewed statistics during the summer and those that didnt review.

Now, if we come back to school and there is a significant difference we would state:Our null hypothesis has been rejected. GET IT??

If the dread of math is setting in, please know it will get better quickly and be open minded. After a little bit of struggle it really is not that bad. I have found all the tech tricks with graphs, etc, actually makes it kinda cool. We will work on this during the entire school year during our investigation time.This material, along with other supporting statistics resources will remain available to help you each time.

We will build our expertise over time,this will not happen overnight.

It really makes things clear.You have probably asked yourself after many labs during your lifetime, What did our results mean?, How do we know if it really has any meaning at all?, How could I test this to see if my experiment had any real meaning?

Make some time for this!!www.randi.org

www.badscience.net

MythBusters

Ted Talks Battling Bad Sciencehttp://www.ted.com/talks/ben_goldacre_battling_bad_science

Vocabularyhttp://www.fiserscience.com/statistics.html I have a whole page of words I think you might need as a reference place.

Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data.

Descriptive statistics consists of the collection, organization, summarization, and presentation of data.

Inferential statistics consists of generalizing from samples to populations, performing estimations and hypothesis tests, determining relationships among variables, and making predictions.

Probability: the chance of the event occurring. Use in inferential statistics.

More vocabulary:A population consists of all subjects (human or otherwise) that are being studied.

A sample is a group of subjects selected from a population.

Discrete variables assume values that can be counted.

Continuous variables can assume an infinite number of values between any two specific values. They are obtained by measuring. They often include fractions and decimals

Qualitative vs. QuantitativeMarital status of nurses in hospital.Time it takes to run a marathon.Weights of lobsters in a tank in a restaurant.Colors of automobiles in a shopping center parking lot.Ounces of ice cream in ice cream float.Capacity of the NFL football stadiums.Ages of people living in a personal care home. Discrete vs. ContinuousNumber of pizzas sold by Pizza Express each day.Relative humidity levels in operating rooms at local hospitals.Number of bananas in a bunch at several local supermarkets.Lifetime (in hours) of 15 Ipod batteries.Weights of the backpacks of first graders on a school bus.Number of students each day who make appointments with a math tutor at a local college.Blood pressures of runners in a marathon.

1. GraphingThe first step in data analysis and data exploration is to create graphical displays of the data that reflect the questions the investigation purports to answer.Effective graphs convey summary or descriptive statistics as part of the display. It is not practical to think you can count everything.Most of the time, that would simply be impossible. So we take samples.Statistics takes samples and make inferencesthat represent the true population. Sampling follows predictable patterns.Its theoretical. Mean, Median, Mode(Central tendencies)Mean means average, so the first time you think Im mean.just know Im average. =)To get the mean:

The average value away from the average.

Math is fun. Good website. http://www.mathsisfun.com/data/standard-deviation.htmlThere are some simple standard deviation problems atThe bottom of the page. CategoriesThe data you collect to will generally fall into three Categories:Parametric: Normal data. Fits normal curve or distribution. Generally in a decimal form. The bell curve! Examples: plant height, body temperature, response rate. (The prefix para means side by side. So close in comparison, not likean outlier. If para means normal, does paranormal activity mean normal normal activity? )Nonparametric: not normal. Does not fit a normal distribution. May include outliers. Can be ordered. Can be big, medium, small (qualitative) Frequency or count data: How many fits in this category? Flies you counted with normal wings, etc. Questions to divide into 2 groups. Is A different than B?How are A and B correlated?

To design a good investigation, the type of data needed to answer your question has to be decided during the planning stages. What kind of question is being asked and what kind of data will be required to answer it?5 types of graphs1. Bar Graphs2. Scatter plots3. Box and whiskers plot4. Histograms5. Line Graphs Investigations that need a graph:1. Artificial Selection AP exam 20144. Diffusion and Osmosis5. Photosynthesis6. Cellular Respiration9. Biotechnology: Restriction Enzyme Analysis of DNA12. Fruit Fly Behavior AP exam 201313. Enzyme Activity Bar GraphsUse two compare two samples of categorical or count data. You will need to compare the calculated means with error bars. Example: qualitative: eye color. Always use Sample Standard error bar. (Sample error of the mean) (SEM)

Bar graph with only the mean plotted.

ScatterplotsUsed to explore associations between two variables visually. One variable is measured against another. It is looking for trends or associations. Plotting of individual data points on an x-y plot.

Scatterplots

Bell-curved

Sine wave-likeBox-and-whisker plotAllows comparison of two samples of nonparametric data (data that does not fit a normal distribution)Medians and quartiles

Lowest value, 2: Q1, 3: the Median; 4. Q3, 5. highest valueOf set. Histograms use this as a preliminary step to see shape. Frequency diagram. Used to display the distribution of data, providing a representation of the central tendencies (mean, Median, Mode) and the spread of the data. Can then see if parametric or non. Involves measurement data and data distribution.Requires setting up bins which are uniform range intervals that cover the entire range of the data. Then the number of measurements that fit in each bin (range of unit) are counted and graphed. If enough measurements are made, the data can show the normal distribution, or bell-shaped distribution, on a histogram.Quantitative, uses averages, spreads from low to high; continuous data such as time, or measurements; tenths show minor changes.

Sample size should be at least 30.

Record High Temperatures

ClassBoundariesFrequency99.5-104.52

104.5-109.58109.5-114.518114.5-119.513119.5-124.57124.5-129.51129.5-134.51Ipad app: Numbers+ Create new spreadsheet: blankEnter just as you see on table.Grab all. Create chart. Add details. Mail symbol. Open in other app then Google Drive. Sunny & Shady Leaves

Histogram - nonparametric

Line GraphsCan use multiple variables.

Usually change over time; line suggests trend

Good Graph reminders:Title

Easy to identify lines and bars

Clearly labeled with units.

X-axis independent variableY-axis dependent variable

Uniform intervalsMore than one condition: different lines:style,colorIndicated on a keyDont start at origin if your data doesnt. Standard Error of the Means bars (SEM)Handbook of Biological StatisticsGood Resource

http://udel.edu/~mcdonald/statintro.html

I find that a systematic, step-by-step approach is the best way to analyze biological data. The statistical analysis of a biological experiment may be broken down into the following steps:Specify the biological question to be answered. Put the question in the form of a biological null hypothesis and alternate hypothesis. Put the question in the form of a statistical null hypothesis and alternate hypothesis. Determine which variables are relevant to the question. Determine what kind of variable each one is. Design an experiment that controls or randomizes the confounding variables. Based on the number of variables, the kind of variables, the expected fit to the parametric assumptions, and the hypothesis to be tested, choose the best statistical test to use. If possible, do a power analysis to determine a good sample size for the experiment. Do the experiment. Examine the data to see if it meets the assumptions of the statistical test you chose (normality, homoscedasticity, etc.). If it doesn't, choose a more appropriate test. Apply the chosen statistical test, and interpret the result. Communicate your results effectively, usually with a graph or table.

Chapter 2: Data AnalysisLabs using DATA ANALYSIS:1. ARTIFICIAL SELECTION FRQ 20142. MATHEMATICAL MODELING: HARDY-WEINBERY3. COMPARING DNA SEQUENCES TO UNDERSTAND EVOLUTIONARY RELATIONSHIPS WITH BLAST4. CELL DIVISION: MITOSIS AND MEIOSIS Data analysisDescriptive statistics: summarize data. Shows variationin the data, standard errors, best-fit functions, and confidence that sufficient data have been collected.

Inferential statistics: involves making conclusions, using experimental data to infer parameters in the natural population. Relies on probability.

Most measurements are continuous, infinite number over a period of time. Examples: size, time, height, weight, absorbance. Others are: Count data: qualitative, categorical, example: number of hairs, etc. Requires different statistical tools. Run through of the steps.Good idea to do this first, before you design your own experiment.

Most investigations will allow to have an initial run-through with all of class doing the same thing.

Components of Data Analysis

BiasHow can we be certain if the statistic we use is giving an accurate estimate of?Think of what we want to estimate (the parameter) as the target and the way we estimate it (the statistic) as the arrow. Our statistic is called unbiased if it is close to being on target. Our statistic is called biased if it is not close to being ontarget.

BiasWe determine bias of a statistic by comparing the center of its sampling distribution to the parameter being estimated.

Sample means and proportions are both unbiased estimators.

For normally distributed populationsWhen a variable in a population is normally distributed, then the sampling distribution of x bar for all possible samples of size n is also normally distributed.

If the population is N(m,s), then the sample means distribution is N(m,s/n).PopulationSample meansSample Standard Error Inference/Confidence Intervals/Empirical Rule/ 67-95-99.7 Rule

Confidence Intervals: Use to determine iftwo populations are different from one another. Think Erika had slide on narrowing due to increase in pop????

Parametric or Nonparametric:maybe if we have time and enough sticky notes, well do this.

Whats in your tool bag?

So, we made a histogram of the information. Looked for our shape. Now concluded parametric or nonparametric.

If parametric, we will use our sample size, mean standard deviation, and standard error to analyze.

If nonparametric, we will use medians, modes, quartiles, and range . Often use a box and whiskers plot.

These are the tools we will use for each type to analyze anddescribe it. Error Bars

Another example: Body Temperature

Explore the sample:

Then descriptive statistics:https://www.khanacademy.org/math/probability/statistics-inferential/confidence-intervals/v/confidence-interval-1

The Khan Academy Video belowwill help youwrap your headaround the all of thesemethods used to analyze.I dont see us needing to do Calculations butAn understandingIs needed. Parametric or Nonparametric?

http://www.washingtonpost.com/blogs/wonkblog/wp/2013/03/06/this-viral-video-is-right-we-need-to-worry-about-wealth-inequality/

Interesting video included in above link about Wealth distribution. Chapter 3: Hypothesis TestingNecessary in the following Investigations:1. Artificial Selection2. Mathematical Modeling: Hardy-Weinberg3. Comparing DNA: BLAST4. Diffusion and Osmosis5. Photosynthesis6. Cellular Respiration7. Cell Division: Mitosis and Meiosis8. Biotechnology: Bacterial Transformation9. Biotechnology: RE of DNA10. Energy Dynamics11. Transpiration12. Fruit-Fly Behavior13. Enzyme ActivityHypothesisA hypothesis is a statement explaining that a causal relationship exists between and underlying factor (variable) and an observable phenomenon. You might propose a tentative explanation and this would be called your working hypothesis.ABSOLUTE PROOF IS NOT POSSIBLE, SO WE FOCUS ON TRYING TO REJECT A NULL HYPOTHESIS.A null hypothesis is a statement that the variable has no causal relationship. An alternative to the null hypothesis would be that there is a size difference, etc. Crucial to understand.We say we reject the null, not that the investigation has proven the alternative hypothesis. A hypothesis is a testable statement explaining some relationship between cause and effect. A prediction is a statement of what youthink will happen. It is good to have more than one working hypotheses. The investigator can remain more open scientifically and not tied to one single explanation. Hypothesis TestingAsks the question:Is there something to these measurements?OrIs the effect real?Hypothesis testingThe data analysis tools used, such as SE, mean, variance, median, etc. are now used to calculate comparative test statistics.

These tests determine a probability and distribution of the samples the null distribution.

A critical value of the probability of whether the results or even more extreme results occur by chance alone, if the null hypothesis is true (probability value, or p-value of less than 5%)

4 possible outcomes of Hypothesis Testing

5% critical values in BiologyDecision point in biology.1% or 0.1% more critical. Like life or death issues in medical studies. The suggested path for including hypothesis testing:1. Choose a statistical test based on the question and types of data. Next page table.2. State the null and alternative hypotheses 3. Design and carry out the investigation.4. Conduct a data analysis, and present graphical and tabular summaries of the data. 5. Carry out the statistical tests. Include the sample size, test statistic chosen, and p-values in the data reports.6. Make a conclusion, always stating the amount of evidence in terms of the alternative hypothesis.

Examples of Hypothesis Testingt test Chi-SquareTrends linear regression and r1. Establish null and alternateH0 null hypothesis There is no difference between the two true populations. H1 alternate hypothesis There is a significant difference between the two population means. Establish critical value p value of 5%Decision table for test P-valueAnalyze p value and critical value. Reject null. Or state results support alternate hypothesis. Do not accept the alternative hypothesis.

T-testExample two groups independent from each other:Unpaired T test : determines how different two sample populations are from one another by comparing means and errors. The College Board manual doesnt go into how to run the test outside of an Excel program, so I dont foresee this as a problem to calculate, maybe just an understanding required. One tail and two tail refer to both sides of the distribution or just one.

The t-test can be used to determine how different two sample populations are from one another by comparing means and standard errors. One sample T-test formula:(sample mean-population mean)The sample standard errorThen the t-test value is taken to a t-test table and you need degrees of freedom and t value to estimate the p value. d= n-1

Comparing two samples and testing for differences1. Construct two histograms to get a visual of distribution, overlap, shape.2. Calculate descriptive statistics.3. Display in bar graphs4. Test for differences. 5. Two sample t-test

Chi-Square ResultsMakes comparison between the data collected and what we expect. We will use this test most often. Genetics, animal behavior.

A p-value of 0.05 means that there is a 5% chance that the difference between the observed and the expected data is a random difference and a 95% chance that the difference is real and repeatable. A significant difference. If the p-value is greater than 0.05 we would fail to reject to the null hypothesis.

This is the table given on your formula sheet. We need towrap your head around the difference of the tables and how the application to the null is still the same. Trends and Associations

Best-fit line, linear regression, and rIs there a general trend?A regression analysis graphs a best fit line that summarizes all the points into a single linear equation.Positive correlation: heart rate goes up when temp goes up. Trend line is created from how far each point is from the mean x value and from the mean y value. The r-value provides an estimate of the degree of correlation. r-value of 1 is positive 1:1 correlationR-value of -1 is negative correlation r-value of .25 equal R2 of 0.064 which means 6.4% of variation (difference) is due to temp and not a strong correlation. R-value does not imply causal relationship.Two tailed t for this since 2 variables. P value on last graph p 0.04. which means 96% confidence that heart rate and body temperature are correlated, although not strongly so. Chapter 4: Mathematical ModelingLabs using Mathematical Modeling1. Artifical Selection 2. Mathematical Modeling: Hardy-Weinberg 3. Comparing DNA: BLAST 9. Biotechnology: RE Analysis of DNA 10. Energy Dynamics11. TranspirationMathematical ModelingThe process of creating mathematical or computer-based representations of the structure and interactions of complex biological systems. What can we measure?What should we measure?What are the relevant variables?What are the simplest informative models that we can build?

Deterministic models are calculated with fixed probabilities.Stochastic models use a random number generator to create a model that has a variable outcome and contingencies, much as normal biological systems do.Example is Investigation 2: Mathematical modeling: Hardy-Weinberg. Example: Simple Model of Population GrowthMaybe we might try this sometime: Building a model.

FRQ writingIf you use statistics to answer the question, use statistics to justify your answer. More resources:http://www.radford.edu/jkell/statsgraphs.pdfhttp://www.biologyforlife.com/graphing.html