top 10 concepts of statistics

7/30/2019 Top 10 concepts of Statistics

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Review of Top 10 Concepts

in Statistics(reordered slightly for review the interactivesession)

NOTE: This Power Point file is not an introduction,but rather a checklist of topics to review


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Top Ten #10

Qualitative vs. Quantitative


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Qualitative

Categorical data:

success vs. failure

ethnicitymarital status

color

zip code4 star hotel in tour guide


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Qualitative

If you need an average, do not calculate themean

However, you can compute the mode(average person is married, buys a blue carmade in America)


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Quantitative

Two cases

Case 1: discrete

Case 2: continuous


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Discrete

(1) integer values (0,1,2,)

(2) example: binomial

(3) finite number of possible values(4) counting

(5) number of brothers

(6) number of cars arriving at gas station


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Continuous

Real numbers, such as decimal values($22.22)

Examples: Z, t Infinite number of possible values

Measurement

Miles per gallon, distance, duration of time


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Graphical Tools

Pie chart or bar chart: qualitative

Joint frequency table: qualitative (relatemarital status vs zip code)

Scatter diagram: quantitative (distance fromCSUN vs duration of time to reach CSUN)


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Hypothesis Testing

Confidence Intervals

Quantitative: Mean

Qualitative: Proportion


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Top Ten #9

Population vs. Sample


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Population

Collection of all items (all light bulbs made atfactory)

Parameter: measure of population

(1) population mean (average number ofhours in life of all bulbs)

(2) population proportion (% of all bulbs thatare defective)


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Sample

Part of population (bulbs tested by inspector)

Statistic: measure of sample = estimate ofparameter

(1) sample mean (average number of hoursin life of bulbs tested by inspector)

(2) sample proportion (% of bulbs in sample

that are defective)


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Top Ten #1

Descriptive Statistics


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Measures of Central Location

Mean

Median

Mode


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Mean

Population mean == x/N = (5+1+6)/3 = 12/3 =4

Algebra: x = N* = 3*4 =12

Sample mean = x-bar = x/n Example: the number of hours spent on the

Internet: 4, 8, and 9

x-bar = (4+8+9)/3 = 7 hours

Do NOT use if the number of observations issmall or with extreme values

Ex: Do NOT use if 3 houses were sold this week,and one was a mansion


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Median

Median = middle value

Example: 5,1,6

Step 1: Sort data: 1,5,6

Step 2: Middle value = 5 When there is an even number of observation,

median is computed by averaging the twoobservations in the middle.

OK even if there are extreme values

Home sales: 100K,200K,900K, so

mean =400K, but median = 200K


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Mode

Mode: most frequent value

Ex: female, male, female

Mode = female Ex: 1,1,2,3,5,8

Mode = 1

It may not be a very good measure, see thefollowing example


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Measures of Central Location -Example

Sample: 0, 0, 5, 7, 8, 9, 12, 14, 22, 23

Sample Mean = x-bar = x/n = 100/10 = 10

Median = (8+9)/2 = 8.5

Mode = 0


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Relationship

Case 1: if probability distribution symmetric(ex. bell-shaped, normal distribution),

Mean = Median = Mode

Case 2: if distribution positively skewed toright (ex. incomes of employers in large firm: a

large number of relatively low-paid workersand a small number of high-paid executives),

Mode < Median < Mean


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Relationship contd

Case 3: if distribution negatively skewed to left(ex. The time taken by students to write

exams: few students hand their exams earlyand majority of students turn in their exam atthe end of exam), Mean < Median < Mode


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Dispersion Measures ofVariability

How much spread of data

How much uncertainty

Measures Range

Variance

Standard deviation


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Range

Range = Max-Min > 0

But range affected by unusual values

Ex: Santa Monica has a high of 105 degreesand a low of 30 once a century, but rangewould be 105-30 = 75


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Standard Deviation (SD)

Better than range because all data used

Population SD = Square root of variance=sigma =

SD > 0


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Empirical Rule

Applies to mound or bell-shaped curves

Ex: normal distribution 68% of data within + one SD of mean

95% of data within + two SD of mean

99.7% of data within + three SD of mean


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Standard Deviation =

Square Root of Variance

1)(2

nxxs


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Sample Standard Deviation

x

6 6-8=-2 (-2)(-2)= 4

6 6-8=-2 4

7 7-8=-1 (-1)(-1)= 1

8 8-8=0 0

13 13-8=5 (5)(5)= 25

Sum=40 Sum=0 Sum = 34

Mean=40/5=8

xx 2)( xx


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Standard Deviation

Total variation = 34

Sample variance = 34/4 = 8.5

Sample standard deviation =square root of 8.5 = 2.9


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Measures of Variability - Example

The hourly wages earned by a sample of five studentsare:

$7, $5, $11, $8, and $6Range: 11 5 = 6

Variance:

Standard deviation:

30.5

15

2.21

15

4.76...4.77

1

222

2

n

XXs

30.230.52

ss


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Graphical Tools

Line chart: trend over time

Scatter diagram: relationship between twovariables

Bar chart: frequency for each category Histogram: frequency for each class of

measured data (graph of frequency distr.)

Box plot: graphical display based onquartiles, which divide data into 4 parts


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Top Ten #8

Variation Creates Uncertainty


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No Variation

Certainty, exact prediction

Standard deviation = 0

Variance = 0

All data exactly same

Example: all workers in minimum wage job


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High Variation

Uncertainty, unpredictable

High standard deviation

Ex #1: Workers in downtown L.A. have variationbetween CEOs and garment workers

Ex #2: New York temperatures in spring rangefrom below freezing to very hot


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Comparing StandardDeviations

Temperature Example

Beach city: small standard deviation (single

temperature reading close to mean) High Desert city: High standard deviation (hot

days, cool nights in spring)


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Standard Error of the Mean

Standard deviation of sample mean =

standard deviation/square root of n

Ex: standard deviation = 10, n =4, so standarderror of the mean = 10/2= 5

Note that 5


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Sampling Distribution

Expected value of sample mean = populationmean, but an individual sample mean could besmaller or larger than the population mean

Population mean is a constant parameter, butsample mean is a random variable

Sampling distribution is distribution of samplemeans


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Example

Mean age of all students in the building ispopulation mean

Each classroom has a sample mean

Distribution of sample means from allclassrooms is sampling distribution


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Central Limit Theorem (CLT)

If population standard deviation is known,sampling distribution of sample means is normalif n > 30

CLT applies even if original population isskewed


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Top Ten #5

Expected Value


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Expected Value

Expected Value = E(x) = xP(x)

= x1P(x1) + x2P(x2) +

Expected value is a weighted average, also along-run average


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Example

Find the expected age at high schoolgraduation if 11 were 17 years old, 80 were18 years old, and 5 were 19 years old

Step 1: 11+80+5=96


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Step 2

x P(x) x P(x)

17 11/96=.115 17(.115)=1.955

18 80/96=.833 18(.833)=14.994

19 5/96=.052 19(.052)=.988

E(x)= 17.937


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Top Ten #4

Linear Regression


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Linear Regression

Regression equation:

=dependent variable=predicted value x= independent variable

b0=y-intercept =predicted value of y if x=0

b1

=slope=regression coefficient

=change in y per unit change in x

xy bb 10

y


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Slope vs Correlation

Positive slope (b1>0): positive correlationbetween x and y (y increase if x increase)

Negative slope (b1


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Simple Linear Regression

Simple: one independent variable, onedependent variable

Linear: graph of regression equation isstraight line


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Example

y = salary (female manager, in thousands ofdollars)

x = number of children

n = number of observations


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Given Data

x y

2 48

1 52

4 33


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Totals

x y

2 48

1 52

4 33 n=3

Sum=7 Sum=133


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Slope (b1) = -6.5

Method of Least Squares formulas not onBUS 302 exam

b1= -6.5 given

Interpretation: If one female manager has 1more child than another, salary is $6,500

lower; that is, salary of female managersis expected to decrease by -6.5 (inthousand of dollars) per child


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Intercept (b0)

33.237

nxx 33.44

3133

nyy

b0 = 44.33 (-6.5)(2.33) = 59.5

If number of children is zero,expected salary is $59,500

xy bb 10


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Regression Equation

xy 5.65.59


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Forecast Salary If 3 Children

59.56.5(3) = 40

$40,000 = expected salary


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xforecasty bb 10

yyerror

2

)(

2

2

n

yy

n

SSES

Standard Error of Estimate


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(1)=x (2)=y (3) =59.5-6.5x

(4)=

(2)-(3)

2 48 46.5 1.5 2.25

1 52 53 -1 1

4 33 33.5 -.5 .25

SSE=3.5

y 2)( yy


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9.15.3

23

5.3

S


Actual salary typically $1,900away from expected salary


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Coefficient of Determination

R2 = % of total variation in y that can beexplained by variation in x

Measure of how close the linear regression

line fits the points in a scatter diagram

R2 = 1: max. possible value: perfect linearrelationship between y and x (straight line)

R2 = 0: min. value: no linear relationship


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Sources of Variation (V)

Total V = Explained V + Unexplained V

SS = Sum of Squares = V

Total SS = Regression SS + Error SS

SST = SSR + SSE

SSR = Explained V, SSE = Unexplained


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R2 =SSRSST

R2 = 197 = .98

200.5

Interpretation: 98% of total variation in salarycan be explained by variation in number of

children


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0 < R2 < 1

0: No linear relationship since SSR=0(explained variation =0)

1: Perfect relationship since SSR = SST

(unexplained variation = SSE = 0), but doesnot prove cause and effect


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R=Correlation Coefficient

Case 1: slope (b1) < 0

R < 0

R is negative square root of coefficient of

determination

2

RR


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Our Example

Slope = b1 = -6.5

R2 = .98

R = -.99


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Case 2: Slope > 0

R is positive square root of coefficient ofdetermination

Ex: R2 = .49

R = .70

R has no interpretation

R overstates relationship


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Caution

Nonlinear relationship (parabola, hyperbola,etc) can NOT be measured by R2

In fact, you could get R2=0 with a nonlinear

graph on a scatter diagram


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Summary: Correlation Coefficient

Case 1: If b1 > 0, R is the positive square rootof the coefficient of determination Ex#1: y = 4+3x, R2=.36: R = +.60

Case 2: If b1 < 0, R is the negative squareroot of the coefficient of determination Ex#2: y = 80-10x, R2=.49: R = -.70

NOTE! Ex#2 has stronger relationship, asmeasured by coefficient of determination


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Extreme Values

R=+1: perfect positive correlation

R= -1: perfect negative correlation

R=0: zero correlation


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MS Excel Output

Correlation Coefficient (-0.9912): Note

that you need to change the sign because

the sign of slope (b1) is negative (-6.5)



Regression Coefficient


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Top Ten #6

What Distribution to Use?


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Use Binomial Distribution If:

Random variable (x) is number of successes in ntrials

Each trial is success or failure

Independent trials

Constant probability of success () on each trial

Sampling with replacement (in practice, people

may use binomial w/o replacement, but theory iswith replacement)


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Success vs. Failure

The binomial experiment can result in onlyone of two possible outcomes:

Male vs. Female

Defective vs. Non-defective Yes or No

Pass (8 or more right answers) vs. Fail (fewer

than 8) Buy drink (21 or over) vs. Cannot buy drink


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Binomial Is Discrete

Integer values

0,1,2,n

Binomial is often skewed, but may be symmetric


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Normal Distribution

Continuous, bell-shaped, symmetric

Mean=median=mode

Measurement (dollars, inches, years)

Cumulative probability under normal curve : useZ table if you know population mean andpopulation standard deviation

Sample mean: use Z table if you know

population standard deviation and either normalpopulation or n > 30


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t Distribution

Continuous, mound-shaped, symmetric

Applications similar to normal

More spread out than normal

Use t if normal population but populationstandard deviation not known

Degrees of freedom = df = n-1 if estimating themean of one population

t approaches z as df increases


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Normal or t Distribution?

Use t table if normal population but populationstandard deviation () is not known

If you are given the sample standard deviation

(s), use t table, assuming normal population


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Top Ten #3

Confidence Intervals: Mean and Proportion


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Confidence Interval

A confidence interval is a range of values withinwhich the population parameter is expectedto occur.


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Factors for Confidence Interval

The factors that determine the width of aconfidence interval are:

1. The sample size, n2. The variability in the population, usually

estimated by standard deviation.

3. The desired level of confidence.


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Confidence Interval: Mean

Use normal distribution (Z table if):

population standard deviation (sigma)known and either (1) or (2):

(1) Normal population

(2) Sample size > 30


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If normal table, then

n

z

n

x


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Normal Table

Tail = .5(1 confidence level)

NOTE! Different statistics texts have differentnormal tables

This review uses the tail of the bell curve

Ex: 95% confidence: tail = .5(1-.95)= .025

Z.025 = 1.96


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Example

n=49, x=490, =2, 95% confidence

9.44 < < 10.56

56.01049

296.1

49

490


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One of SOM professors wants toestimate the mean number of hoursworked per week by students. A sample

of 49 students showed a mean of 24hours. It is assumed that the populationstandard deviation is 4 hours. What isthe population mean?

Another Example


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95 percent confidence interval for thepopulation mean.

12.100.24

49

4

96.100.2496.1

nX

The confidence limits range from 22.88 to

25.12. We estimate with 95 percentconfidence that the average number of hoursworked per week by students lies between

these two values.

Another Example contd



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Confidence Interval: Meant distribution

Use if normal population but populationstandard deviation () not known

If you are given the sample standarddeviation (s), use t table, assuming normalpopulation

If one population, n-1 degrees of freedom



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n

s

n

xtn 1

Confidence Interval: Meant distribution

Confidence Interval:


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Confidence Interval:Proportion

Use if success or failure

(ex: defective or not-defective,

satisfactory or unsatisfactory)

Normal approximation to binomial ok if(n)() > 5 and (n)(1-) > 5, where

n = sample size

= population proportion

NOTE: NEVER use the t table if proportion!!



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Ex: 8 defectives out of 100, so p = .08 and

n = 100, 95% confidence

n

ppzp

)1(

05.08.100

)92)(.08.0(96.108.



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A sample of 500 people who own their houserevealed that 175 planned to sell their homeswithin five years. Develop a 98% confidence

interval for the proportion of people who plan tosell their house within five years.

0497.35.500

)65)(.35(.33.235.

35.0500

175p


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Interpretation

If 95% confidence, then 95% of all confidenceintervals will include the true population parameter

NOTE! Never use the term probability when

estimating a parameter!! (ex: Do NOT sayProbability that population mean is between 23 and32 is .95 because parameter is not a randomvariable. In fact, the population mean is a fixed but

unknown quantity.)


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Point vs Interval Estimate

Point estimate: statistic (single number)

Ex: sample mean, sample proportion

Each sample gives different point estimate

Interval estimate: range of values

Ex: Population mean = sample mean + error

Parameter = statistic + error


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Width of Interval

Ex: sample mean =23, error = 3

Point estimate = 23

Interval estimate = 23 + 3, or (20,26)

Width of interval = 26-20 = 6

Wide interval: Point estimate unreliable


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Wide Confidence Interval If

(1) small sample size(n)

(2) large standard deviation

(3) high confidence interval (ex: 99% confidenceinterval wider than 95% confidence interval)

If you want narrow interval, you need a largesample size or small standard deviation or low

confidence level.


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Top Ten #7

P-value


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P-value

P-value = probability of getting a sample statisticas extreme (or more extreme) than the samplestatistic you got from your sample, given that the

null hypothesis is true


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P-value Example: one tail test

H0: = 40

HA: > 40

Sample mean = 43

P-value = P(sample mean > 43, given H0 true)

Meaning: probability of observing a samplemean as large as 43 when the population mean

is 40 How to use it: Reject H0 if p-value <

(significance level)


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Two Cases

Suppose = .05

Case 1: suppose p-value = .02, then reject H0(unlikely H0 is true; you believe population mean> 40)

Case 2: suppose p-value = .08, then do notreject H0 (H0 may be true; you have reason tobelieve that the population mean may be 40)


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P-value Example: two tail test

H0 : = 70

HA: 70

Sample mean = 72

If two-tails, then P-value =

2 P(sample mean > 72)=2(.04)=.08

If = .05, p-value > , so do not reject H0


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Top Ten #2

Hypothesis Testing


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Population mean=

Population proportion=

A statement about the value of a populationparameter

Never include sample statistic (such as, x-bar) in hypothesis

H0: Null Hypothesis

H H Alt ti H th i


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HA or H1:Alternative Hypothesis

ONE TAIL ALTERNATIVE

Right tail: >number(smog ck)

>fraction(%defectives)

Left tail:


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One-Tailed Tests

A test is one-tailed when the alternatehypothesis, H1 or HA, states a direction, such as:

H1: The mean yearly salaries earned by full-timeemployees is more than $45,000. (>$45,000)

H1: The average speed of cars traveling onfreeway is less than 75 miles per hour. (


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Two-Tail Alternative

Population mean not equal to number (toohot or too cold)

Population proportion not equal to fraction (%

alcohol too weak or too strong)

Two-Tailed Tests


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Two Tailed Tests

A test is two-tailed when no direction isspecified in the alternate hypothesis

H1: The mean amount of time spent for the

Internet is not equal to 5 hours. ( 5).

H1: The mean price for a gallon of gasoline

is not equal to $2.54. ( $2.54).


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Reject Null Hypothesis (H0) If

Absolute value of test statistic* > critical value*

Reject H0 if |Z Value| > critical Z

Reject H0 if | t Value| > critical t

Reject H0 if p-value < significance level (alpha) Note that direction of inequality is reversed!

Reject H0 if very large difference between samplestatistic and population parameter in H

0

* Test statistic: A value, determined from sample information, used to determinewhether or not to reject the null hypothesis.

* Critical value: The dividing point between the region where the null hypothesis isrejected and the region where it is not rejected.


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Example: Smog Check

H0 : = 80

HA: > 80

If test statistic =2.2 and critical value = 1.96,

reject H0, and conclude that the populationmean is likely > 80

If test statistic = 1.6 and critical value = 1.96,

do not reject H0, and reserve judgment aboutH0


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Type I vs Type II Error

Alpha= = P(type I error) = Significance level =probability that you reject true null hypothesis

Beta= = P(type II error) = probability you do notreject a null hypothesis, given H0 false

Ex: H0 : Defendant innocent

= P(jury convicts innocent person)

=P(jury acquits guilty person)


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Type I vs Type II Error

H0 true H0 false

Reject H0 Alpha = =P(type I error)

1 (CorrectDecision)

Do not reject H0 1 (CorrectDecision) Beta = =P(type II error)

E l S Ch k


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Example: Smog Check

H0 : = 80

HA: > 80

If p-value = 0.01 and alpha = 0.05, reject H0,

and conclude that the population mean islikely > 80

If p-value = 0.07 and alpha = 0.05, do not

reject H0, and reserve judgment about H0

Test Statistic


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Test Statistic

When testing for the population mean from alarge sample and the population standarddeviation is known, the test statistic is given

by:

zX

/ n

E l


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The processors of Best Mayo indicate on thelabel that the bottle contains 16 ounces ofmayo. The standard deviation of the process

is 0.5 ounces. A sample of 36 bottles from lasthours production showed a mean weight of16.12 ounces per bottle. At the .05significance level, can we conclude that themean amount per bottle is greater than 16ounces?

Example

E l td


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1. State the null and the alternative hypotheses:H0: = 16, H1: > 16

3. Identify the test statistic. Because we know thepopulation standard deviation, the test statistic is z.

4. State the decision rule.

Reject H0 if |z|>1.645 (= z0.05)

2. Select the level of significance. In this case,

we selected the .05 significance level.

Example contd

E l td


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5. Compute the value of the test statistic

44.1

365.0

00.1612.16

n

Xz

6. Conclusion: Do not reject the null hypothesis.

We cannot conclude the mean is greater than 16ounces.

Example contd

top 10 concepts of statistics

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