Hypothesis Testing

11

Portions of these notes are adapted from Statistics 7e © 1997 Prentice-

Hall, Inc.

What is a testable What is a testable hypothesis, anyway?hypothesis, anyway?

Tools for Samples ContinuedTools for Samples Continued

Inferences Based on a Single Sample: Inferences Based on a Single Sample: Tests of HypothesisTests of Hypothesis

PBAF 527 Winter 2005PBAF 527 Winter 2005

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Hall, Inc.TodayToday

1.1. Tools for Samples (recap)Tools for Samples (recap)

2.2. Confidence Intervals and Sample Size: Scallops, Sampling, and the Confidence Intervals and Sample Size: Scallops, Sampling, and the LawLaw

3.3. Hypothesis TestingHypothesis Testing

1.1. Distinguish Types of Hypotheses Distinguish Types of Hypotheses

2.2. Describe Hypothesis Testing ProcessDescribe Hypothesis Testing Process

3.3. Solve Hypothesis Testing Problems Based on a Single SampleSolve Hypothesis Testing Problems Based on a Single Sample

4.4. Explain Power of a TestExplain Power of a Test

5.5. Explain p-Value ConceptExplain p-Value Concept

6.6. Summarize the Elements of a Hypothesis TestSummarize the Elements of a Hypothesis Test

7.7. Things to Think about When Hypothesis TestingThings to Think about When Hypothesis Testing

8.8. Special CasesSpecial Cases Small Sample Test Small Sample Test Large Sample Test of ProportionsLarge Sample Test of Proportions

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Hall, Inc.Tools for SamplesTools for Samples

Point EstimatePoint Estimate Best guess of a population parameter based upon a Best guess of a population parameter based upon a

sample sample

Confidence IntervalConfidence Interval Range estimate around point estimateRange estimate around point estimate

Hypothesis TestHypothesis Test Decision rule for rejecting hypothesized population Decision rule for rejecting hypothesized population

values (null hypotheses)values (null hypotheses)

p-valuep-value Continuous measure of support for null hypothesis (a Continuous measure of support for null hypothesis (a

probability, probability, ))

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Hall, Inc.Tools for SamplesTools for Samples

Point EstimatePoint Estimate Best guess of a population parameter based upon a Best guess of a population parameter based upon a

sample sample

Confidence IntervalConfidence Interval Range estimate around point estimateRange estimate around point estimate

Hypothesis TestHypothesis Test Decision rule for rejecting hypothesized population Decision rule for rejecting hypothesized population

values (null hypotheses)values (null hypotheses)

p-valuep-value Continuous measure of support for null hypothesis (a Continuous measure of support for null hypothesis (a

probability, probability, ))

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Hall, Inc.Statistical MethodsStatistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

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Hall, Inc.Hypothesis TestingHypothesis Testing

77



PopulationPopulation

88




I believe the population mean age is 50 (hypothesis).

99





MeanMean X X = 20= 20

Random Random samplesample

1010





MeanMean X X = 20= 20

Reject hypothesis! Not close.

Reject hypothesis! Not close.

Random Random samplesample

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What’s a What’s a Hypothesis?Hypothesis?

1.1. A Belief about a A Belief about a Population ParameterPopulation Parameter

Parameter Is Parameter Is PopulationPopulation Mean, Mean, Proportion, VarianceProportion, Variance

Must Be StatedMust Be StatedBeforeBefore Analysis Analysis

I believe the mean GPA I believe the mean GPA of this class is 3.5!of this class is 3.5!

© 1984-1994 T/Maker Co.

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Hall, Inc.A question, oftenA question, often

What is the relationship between being a What is the relationship between being a women and living in poverty?women and living in poverty?

How fast is the response time to a new How fast is the response time to a new drug?drug?

Is the level of pollutant emitted by an Is the level of pollutant emitted by an industrial facility too high?industrial facility too high?

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A more specific A more specific question!question!

• Do single female householders in Do single female householders in Washington State make, on average, Washington State make, on average, the poverty rate for a family of three?the poverty rate for a family of three?

• Is the time to respond to a new drug Is the time to respond to a new drug 1.2 seconds?1.2 seconds?

• Is the level of pollutant emitted by an Is the level of pollutant emitted by an industrial facility in violation of the EPA industrial facility in violation of the EPA standard of 55 ppm?standard of 55 ppm?

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Construct Construct HypothesesHypotheses

In the case of the drug trial:In the case of the drug trial:

Null hypothesis: Null hypothesis: = 1.2 = 1.2 This we hold to be true until falsifiedThis we hold to be true until falsified

Alternative hypotheses: Alternative hypotheses: 1.2 1.2 Everything but the null hypothesisEverything but the null hypothesis

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Hall, Inc.Null HypothesisNull Hypothesis

1.1. What Is TestedWhat Is Tested

2.2. Has Serious Outcome If Incorrect Decision Has Serious Outcome If Incorrect Decision MadeMade

3.3. Always Has Equality Sign: Always Has Equality Sign: , , , or , or 4.4. Designated HDesignated H00

5.5. Specified as HSpecified as H00: : Some Numeric Value Some Numeric Value Specified with = Sign Even if Specified with = Sign Even if , or , or Example, HExample, H00: : 1.2 1.2

IsIsxx close to this hypothesized value? close to this hypothesized value?

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Alternative Alternative HypothesisHypothesis

1.1. Opposite of Null HypothesisOpposite of Null Hypothesis

2.2. Always Has Inequality Sign:Always Has Inequality Sign: ,,, or , or

3.3. Designated HDesignated Haa

4.4. Specified HSpecified Haa: : < Some Value < Some Value Example, HExample, Haa: : < 3 < 3

Everything but the value of the null hypothesis.Everything but the value of the null hypothesis.

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Hall, Inc.A hypothesis test…A hypothesis test…

• Is a decision rule for rejecting or failing Is a decision rule for rejecting or failing to reject the null hypothesis.to reject the null hypothesis.

• Sets significance level of cut-off Sets significance level of cut-off Often 5% or 1%Often 5% or 1%

• Will reject null if not within CI at same Will reject null if not within CI at same confidence levelconfidence level

• Will reject null if p is less than Will reject null if p is less than

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Hall, Inc.Basic IdeaBasic Idea

1919



S am p le M ea n = 50 S am p le M ea n = 50HH00HH00

Sampling DistributionSampling Distribution

2020



S am p le M ea n = 50 S am p le M ea n = 50


It is unlikely It is unlikely that we would that we would get a sample get a sample mean of this mean of this value ...value ...

20202020HH00HH00

2121






... if in fact this were... if in fact this were the population mean the population mean

20202020HH00HH00

2222






... if in fact this were... if in fact this were the population mean the population mean

... therefore, ... therefore, we reject the we reject the hypothesis hypothesis

that that = 50.= 50.

20202020HH00HH00

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2-Tailed Test About a 2-Tailed Test About a MeanMean

Is the observed value the same as the true value?Is the observed value the same as the true value?

Is the time to respond to a new drug 1.2 Is the time to respond to a new drug 1.2 seconds? seconds?

Suppose 100 rats were injected with the drug. Suppose 100 rats were injected with the drug. The mean response time at the end of the The mean response time at the end of the experiments was 1.05 seconds, and the standard experiments was 1.05 seconds, and the standard deviation .5 seconds.deviation .5 seconds.

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1.1. Establish hypotheses Establish hypotheses HH00::= 1.2 H= 1.2 Haa: : 1.21.2

2.2. Set the decision rule for the test: Set the decision rule for the test: if |z|>zif |z|>z/2/2 then reject the null then reject the null

hypothesis.hypothesis. Draw a picture. Draw a picture.

• pick pick =.05 (for two-sided test this is .025 in each tail)=.05 (for two-sided test this is .025 in each tail)

• find zfind z/2/2zz/2/2=1.96=1.96

3.3. Find z-statistic (like z-score)Find z-statistic (like z-score)

4.4. Compare test statistic to critical value.Compare test statistic to critical value.

Since |z|> zSince |z|> z/2/2 we can reject the null hypothesis at a 5% level. we can reject the null hypothesis at a 5% level.

The response time is not actually 1.2 seconds.The response time is not actually 1.2 seconds.

0.305.

15.

100

5.2.105.1

n

xxz

x

0.305.

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5.2.105.1

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x

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Is the observed value greater then or less Is the observed value greater then or less than the true value?than the true value?

The need for action determines the The need for action determines the alternative hypothesis. Think of this as alternative hypothesis. Think of this as the research hypothesis.the research hypothesis.

Is the sample mean response time of 1.05 Is the sample mean response time of 1.05 seconds seconds lower thanlower than 1.2 seconds? 1.2 seconds?

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1-Tailed Test About a 1-Tailed Test About a Mean:Mean:

Left-Tailed TestLeft-Tailed Test1. Establish hypotheses 1. Establish hypotheses HH00::1.2 H1.2 Haa: : <1.2<1.2

TheThe need for action is if the observed value is less than 1.2, so that’s need for action is if the observed value is less than 1.2, so that’s the alternative hypothesis.the alternative hypothesis.

2. Set the decision rule for the test: 2. Set the decision rule for the test: if z<zif z<z then reject the null hypothesis. then reject the null hypothesis. Draw a picture.Draw a picture.

• pick pick =.05 (for one-sided test this is .5 in the tail)=.05 (for one-sided test this is .5 in the tail)• find zfind z zz=-1.645=-1.645

3.3. Find test statisticFind test statistic

4. Compare test statistic to critical value.4. Compare test statistic to critical value.

Since z< zSince z< z we can reject the null hypothesis at a 5% level. The we can reject the null hypothesis at a 5% level. The response time actually less than 1.2 seconds.response time actually less than 1.2 seconds.

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Right-Tailed TestRight-Tailed TestThe building specifications in a certain city require that the The building specifications in a certain city require that the average breaking strength of residential sewer pipe be average breaking strength of residential sewer pipe be more than 2,400 pounds per foot of length. Each more than 2,400 pounds per foot of length. Each manufacturer who wants to sell pipe in this city must manufacturer who wants to sell pipe in this city must demonstrate that its product meets the specification. demonstrate that its product meets the specification.

So, we want to decide whether the mean breaking strength So, we want to decide whether the mean breaking strength of the pipe exceeds 2,400 pounds per linear foot. of the pipe exceeds 2,400 pounds per linear foot.

We tested 50 sections of pipe and found the mean and We tested 50 sections of pipe and found the mean and standard deviations of the 50 measurements to be:standard deviations of the 50 measurements to be:

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Right-Tailed TestRight-Tailed Test = 2,460 lbs for linear foot= 2,460 lbs for linear foot s=200 lbs per linear foots=200 lbs per linear foot

1.1. Establish hypotheses Establish hypotheses

2. Set the decision rule for the test: if z>z2. Set the decision rule for the test: if z>z then reject the then reject the null hypothesis. Draw a picture.null hypothesis. Draw a picture.

Pick Pick

Find zFind z



H 0:2,400 Ha: >2,400

xx

=.05 (for one-sided test this is .05 in the tail)

z=1.645

Since z> z we can reject the null hypothesis at a 5% level. The company’s pipe has a mean strength that exceeds 2,400 pounds per linear foot.

12.228.28

60

50

200400,2460,2

n

xxz

x

12.2

28.28

60

50

200400,2460,2

n

xxz

x

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Rejection Regions Rejection Regions (Two-Tailed Test) (Two-Tailed Test)

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HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion


1 - 1 -

Level of ConfidenceLevel of Confidence

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HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion


1 - 1 -


Observed sample statisticObserved sample statistic

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HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion


1 - 1 -


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HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion


1 - 1 -


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Rejection Region Rejection Region (One-Tail Test) (One-Tail Test)

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HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion


1 - 1 -


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HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion


1 - 1 -


Observed sample statisticObserved sample statistic

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HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion


1 - 1 -


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Rejection Regions

Summarized

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Errors in Errors in Making DecisionMaking Decision

1.1. Type I ErrorType I Error Probability of Rejecting True Null HypothesisProbability of Rejecting True Null Hypothesis Has Serious ConsequencesHas Serious Consequences Probability of Type I Error Is Probability of Type I Error Is (Alpha)(Alpha)

Called Level of SignificanceCalled Level of Significance

2.2. Type II ErrorType II Error Probability of Failing to Reject a False Null Probability of Failing to Reject a False Null

HypothesisHypothesis Probability of Type II Error Is Probability of Type II Error Is (Beta)(Beta)

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Jury Trial H0 Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0

False

Innocent Correct ErrorDo NotReject

H0

1 - Type IIError

()

Guilty Error Correct RejectH0

Type IError ()

Power(1 - )

Jury Trial H0 Test



False

Innocent Correct ErrorDo NotReject

H0

1 - Type IIError

()


Type IError ()

Power(1 - )

Decision ResultsDecision Results

HH00: Innocent: Innocent

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Jury Trial H0 Test



False

Innocent Correct Error AcceptH0

1 - Type IIError

()


Type IError ()

Power(1 - )

Jury Trial H0 Test



False

Innocent Correct Error AcceptH0

1 - Type IIError

()


Type IError ()

Power(1 - )

Decision ResultsDecision Results

HH00: Innocent: Innocent

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Hall, Inc.

& & Have an Have an Inverse RelationshipInverse Relationship

You can’t reduce both errors simultaneously!

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Hall, Inc.

Pipe Example, Pipe Example, RevisitedRevisited

= 2,430 lbs for linear foot= 2,430 lbs for linear foot s=200 lbs per linear foots=200 lbs per linear foot

1.1. Establish hypotheses Establish hypotheses

2. Set the decision rule for the test: if z>z2. Set the decision rule for the test: if z>z then reject the null then reject the null hypothesis. Draw a picture.hypothesis. Draw a picture.

• Pick Pick

• Find zFind z

3. Find test statistic3. Find test statistic


H 0:2,400 Ha: >2,400 xx


z=1.645

Since z> z is NOT TRUE we cannot reject the null hypothesis at a 5% level. The mean is larger, but does not exceed the specification by enough to provide convincing evidence that the population mean exceeds 2,400.

06.128.28

30

50

200400,2430,2

n

xxz

x

06.1

28.28

30

50

200400,2430,2

n

xxz

x

What if the sample mean breaking strength for the pipe turned out What if the sample mean breaking strength for the pipe turned out to be 2,430 pounds, but the sample standard deviation remains to be 2,430 pounds, but the sample standard deviation remains 200 pounds per linear foot. 200 pounds per linear foot.

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Hall, Inc.Level of SignificanceLevel of Significance

1.1. ProbabilityProbability

2.2. Defines Unlikely Values of Sample Statistic Defines Unlikely Values of Sample Statistic if Null Hypothesis Is Trueif Null Hypothesis Is True Called Rejection Region of Sampling Called Rejection Region of Sampling

DistributionDistribution

3.3. Designated Designated (alpha)(alpha) Typical Values Are .01, .05, .10Typical Values Are .01, .05, .10

4.4. Selected by Researcher at StartSelected by Researcher at Start

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Hall, Inc.P-ValueP-Value

• Probability that you’d get a sample value this Probability that you’d get a sample value this far from the mean or more far from the mean or more IFIF the null the null hypothesis were true. hypothesis were true.

• A continuous measure of “strength” of A continuous measure of “strength” of evidence for null hypothesisevidence for null hypothesis

• Higher p value means less evidence that Ha Higher p value means less evidence that Ha is right (more support for Ho)is right (more support for Ho)

• 1-sided p values give value for one end of the 1-sided p values give value for one end of the distribution; 2 sided includes both.distribution; 2 sided includes both.

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Hall, Inc.

Is z=15 the same as Is z=15 the same as z=2?z=2?

S am p le M ea n = 50 S am p le M ea n = 50 15151515

HH00HH00

µµ 22

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Hall, Inc.P-Value ExampleP-Value Example

What is the probability of getting an observed What is the probability of getting an observed mean of 1.05 seconds to respond to the drug mean of 1.05 seconds to respond to the drug if the true mean is 1.2 seconds? There were if the true mean is 1.2 seconds? There were 100 rats were injected with the drug, the 100 rats were injected with the drug, the mean response time at the end of the mean response time at the end of the experiments was 1.05 seconds, and the experiments was 1.05 seconds, and the standard deviation was .5 seconds.standard deviation was .5 seconds.

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Hall, Inc.P-Value ExampleP-Value Example

To find P value:To find P value:

1. Set up hypotheses: 1. Set up hypotheses: HH00::= 1.2 H= 1.2 Haa: : 1.21.2

2. Find estimate of mean and standard error:2. Find estimate of mean and standard error:

3. Find test statistic (z score):3. Find test statistic (z score):

4. Find p value (probability) 4. Find p value (probability) associated with test statistic from t or z table (large samples, associated with test statistic from t or z table (large samples, use z-table)use z-table)

p=P(|z|>3|Hp=P(|z|>3|H00 true)=2(.0013)=.0026 is the probability that you’d true)=2(.0013)=.0026 is the probability that you’d

get a value of 1.05 if the actual mean were 1.2.get a value of 1.05 if the actual mean were 1.2.

05.1x 05.1x

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xz

0.3

05.

2.105.1

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xz

05.100

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n

sx

05.100

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sx

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Summary: Summary: Elements of a Hypothesis Elements of a Hypothesis

Test (1)Test (1)

Null Hypothesis (H0)Null Hypothesis (H0) A theory about the values of one or more population A theory about the values of one or more population

parameters. The status quo.parameters. The status quo.

Alternative Hypothesis (Ha) Alternative Hypothesis (Ha) A theory that contradicts the null hypothesis. The theory A theory that contradicts the null hypothesis. The theory

generally represents that which we will accept only when generally represents that which we will accept only when sufficient evidence exists to establish its truth.sufficient evidence exists to establish its truth.

Test Statistic Test Statistic A sample statistic used to decide whether to reject the null A sample statistic used to decide whether to reject the null

hypothesis. In general,hypothesis. In general,

Error Standard

Parameter edHypothesiz-Estimatestatistictest

Error Standard

Parameter edHypothesiz-Estimatestatistictest

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Summary: Summary: Elements of a Hypothesis Elements of a Hypothesis

Test (2)Test (2)

Critical ValueCritical Value A value to which the test statistic is compared at some particular A value to which the test statistic is compared at some particular

significance level. (usually apha=.01, .05, .10)significance level. (usually apha=.01, .05, .10)

Rejection RegionRejection Region The numerical values of the test statistic for which the null The numerical values of the test statistic for which the null

hypothesis will be rejected. hypothesis will be rejected. The probability is The probability is that the rejection region will contain the test that the rejection region will contain the test

statistic when the null hypothesis is true, leading to a Type I error. statistic when the null hypothesis is true, leading to a Type I error. is usually chosen to be small (.01, .05, .10) and is the level of is usually chosen to be small (.01, .05, .10) and is the level of significance of the test.significance of the test.

Experiment and calculation of test statisticExperiment and calculation of test statistic Sample from the population and determine the numerical value of Sample from the population and determine the numerical value of

the test statistic.the test statistic.

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Some things to think Some things to think aboutabout

1.1. What happens when n is large or What happens when n is large or σσ is small? is small?

2.2. When can we automatically accept the null When can we automatically accept the null hypothesis?hypothesis?

3.3. What if we misspecify the null hypothesis?What if we misspecify the null hypothesis?

4.4. Practical significance is not the same thing Practical significance is not the same thing as statistical significance.as statistical significance.

5.5. Always formulate HAlways formulate H0 0 and Hand Haa BEFORE you BEFORE you

analyze the data. No snooping!analyze the data. No snooping!

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Hypothesis Testing Hypothesis Testing When When

n is Small and n is Small and σσ UnknownUnknown

Because the sample is smallBecause the sample is small Cannot assume normalityCannot assume normality Cannot assume s is a good approximation for Cannot assume s is a good approximation for σσ

So, use t-distribution:So, use t-distribution:

with n-1 degrees of freedomwith n-1 degrees of freedom

n

sx

t

n

sx

t

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Small Sample t-test Small Sample t-test Example 1 (1)Example 1 (1)

Most water treatment facilities monitor the quality of their drinking Most water treatment facilities monitor the quality of their drinking water on hourly basis. One variable monitored it is pH, which water on hourly basis. One variable monitored it is pH, which measures the degree of alkalinity or acidity in the water. A pH measures the degree of alkalinity or acidity in the water. A pH below 7.0 is acidic, one above 7.0 is alkaline, and a pH of 7.0 is below 7.0 is acidic, one above 7.0 is alkaline, and a pH of 7.0 is neutral. One water treatment plant has a target pH of 8.5 (most neutral. One water treatment plant has a target pH of 8.5 (most try to maintain a slightly alkaline level). The mean and standard try to maintain a slightly alkaline level). The mean and standard deviation of 1 hour’s test results, based on 17 water samples at deviation of 1 hour’s test results, based on 17 water samples at this plant are:this plant are:

Does this sample provide sufficient evidence that the mean pH Does this sample provide sufficient evidence that the mean pH level in the water differs from 8.5?level in the water differs from 8.5?

s=.16s=.1624.8x 24.8x

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1. Establish hypotheses1. Establish hypotheses

2. Set the decision rule for the test: 2. Set the decision rule for the test:

pick pick find tfind t at n-1 dfat n-1 df



05.2039.

08.

17

16.5.842.8

n

sx

t 05.2

039.

08.

17

16.5.842.8

n

sx

t

=.05 (for two-sided test this is .025 in each tail)

H 0:=8.5 Ha: 8.5

if |t|>t at n-1 df then reject the null hypothesis

t=2.12 with 16 degrees of freedom

Since |t|< t we cannot the null hypothesis at a 5% level. We cannot conclude that that the mean pH differs from the target based on the sample evidence.

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A major car manufacturer wants to test a new engine to A major car manufacturer wants to test a new engine to determine whether it meets new air-pollution standards. The determine whether it meets new air-pollution standards. The mean emission mean emission for all engines of this type must be less than 20 for all engines of this type must be less than 20 parts per million of carbon. 10 engines are manufactured for parts per million of carbon. 10 engines are manufactured for testing purposes, and the emission level for each is determined. testing purposes, and the emission level for each is determined. The mean and standard deviation for the tests are:The mean and standard deviation for the tests are:

Do the data supply enough evidence to allow the manufacturer to Do the data supply enough evidence to allow the manufacturer to conclude that this type of engine meets the pollution standard? conclude that this type of engine meets the pollution standard? Assume the manufacturer is willing to risk a Type I error with Assume the manufacturer is willing to risk a Type I error with probability probability =.01.=.01.

s=2.98s=2.9817.17x 17.17x

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pick pick find tfind t



00.3

10

98.22017.17

n

sx

t 00.3

10

98.22017.17

n

sx

t


H 0:≥20 Ha: <20

if t<t then reject the null hypothesis

t=-2.821 with 9 degrees of freedom

We can reject the null. The actual value is less than 20 ppm, and the new engine type meets the pollution standard.

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Hall, Inc.

Large Sample Test for Large Sample Test for the Population the Population

ProportionProportion

When the sample size is large (np and nq When the sample size is large (np and nq are greater than 5)are greater than 5) Assume is distributed normally with Assume is distributed normally with

mean p and standard deviationmean p and standard deviation where q=1-pwhere q=1-p

Test statistic: Test statistic:

2- or 1-tailed tests2- or 1-tailed tests

p̂p̂

n

pqn

pq

nqp

ppz

00

0ˆ

nqp

ppz

00

0ˆ

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Large Sample Tests Large Sample Tests for Proportion for Proportion Example (1)Example (1)

In screening women for breast cancer, doctors use a In screening women for breast cancer, doctors use a method that fails to detect cancer in 20% of the women method that fails to detect cancer in 20% of the women who actually have the disease. Suppose a new method who actually have the disease. Suppose a new method has been developed that researchers hope will detect has been developed that researchers hope will detect cancer more accurately. This new method was used to cancer more accurately. This new method was used to screen a random sample of 140 women known to have screen a random sample of 140 women known to have breast cancer. Of these, the new method failed to detect breast cancer. Of these, the new method failed to detect cancer in 12 women. cancer in 12 women.

Does this sample provide evidence that the failure rate Does this sample provide evidence that the failure rate of the new method differs from the one currently in use?of the new method differs from the one currently in use?

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Large Sample Tests Large Sample Tests for Proportion for Proportion Example (2)Example (2)



pick pick find tfind t



=.05 (for two-sided test this is .025 in each tail)

H 0:p=.2 Ha: p≠.2

if |z|>z then reject the null hypothesis

z=1.96 36.3

034.

114.

140)8)(.2(.

2.086.ˆ

00

0

nqp

ppz 36.3

034.

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2.086.ˆ

00

0

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Since the test statistic falls in the rejection region, we can reject the null. The rate of detection for the new test differs from the old at a .05 level of significance.

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It helps to It helps to remember…remember…

““Hypothesis testing is a ritualized exercise Hypothesis testing is a ritualized exercise in devil’s advocacy.”in devil’s advocacy.”

---Robert P. ---Robert P. AbelsonAbelson

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Scallops, Sampling, Scallops, Sampling, and the Lawand the Law

• Read CaseRead Casea.a. Can a reliable estimate of the mean weight of all Can a reliable estimate of the mean weight of all

the scallops be obtained from a sample size of the scallops be obtained from a sample size of 18?18?

b.b. Do you see any flaws in the rule to confiscate a Do you see any flaws in the rule to confiscate a scallop catch if the sample mean weight is less scallop catch if the sample mean weight is less than 1/36 of a pound?than 1/36 of a pound?

c.c. Develop your own procedure for determining Develop your own procedure for determining whether a ship is in violation of the weight whether a ship is in violation of the weight restriction.restriction.

d.d. Apply your procedure to the data provided.Apply your procedure to the data provided.

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1.1. Tools for Samples (recap)Tools for Samples (recap)

2.2. Confidence Intervals and Sample Size: Scallops, Sampling, and the Confidence Intervals and Sample Size: Scallops, Sampling, and the LawLaw

3.3. Hypothesis TestingHypothesis Testing

1.1. Distinguish Types of Hypotheses Distinguish Types of Hypotheses

2.2. Describe Hypothesis Testing ProcessDescribe Hypothesis Testing Process

3.3. Solve Hypothesis Testing Problems Based on a Single SampleSolve Hypothesis Testing Problems Based on a Single Sample

4.4. Explain Power of a TestExplain Power of a Test

5.5. Explain p-Value ConceptExplain p-Value Concept

6.6. Summarize the Elements of a Hypothesis TestSummarize the Elements of a Hypothesis Test

7.7. Things to Think about When Hypothesis TestingThings to Think about When Hypothesis Testing

8.8. Special CasesSpecial Cases Small Sample Test Small Sample Test Large Sample Test of ProportionsLarge Sample Test of Proportions

End of Chapter

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Education

Hypothesis Testing