Slides by JOHN LOUCKS St. Edward’s University

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© 2009 Thomson South-Western. All Rights Reserved© 2009 Thomson South-Western. All Rights Reserved

Slides by

JOHNLOUCKSSt. Edward’sUniversity

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Chapter 9Chapter 9 Hypothesis Testing Hypothesis Testing

Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses Type I and Type II ErrorsType I and Type II Errors Population Mean: Population Mean: Known Known Population Mean: Population Mean: Unknown Unknown Population ProportionPopulation Proportion

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Developing Null and Alternative Developing Null and Alternative HypothesesHypotheses

Hypothesis testingHypothesis testing can be used to determine whether can be used to determine whether a statement about the value of a population parametera statement about the value of a population parameter should or should not be rejected.should or should not be rejected. The The null hypothesisnull hypothesis, , denoted by denoted by HH0 0 , , is a tentativeis a tentative assumption about a population parameter.assumption about a population parameter. The The alternative hypothesisalternative hypothesis, denoted by , denoted by HHaa, is the, is the

opposite of what is stated in the null hypothesis.opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test isThe alternative hypothesis is what the test is attempting to establish.attempting to establish.

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Testing Research HypothesesTesting Research Hypotheses


• The research hypothesis should be expressed asThe research hypothesis should be expressed as the alternative hypothesis.the alternative hypothesis.

• The conclusion that the research hypothesis is trueThe conclusion that the research hypothesis is true comes from sample data that contradict the nullcomes from sample data that contradict the null hypothesis.hypothesis.

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Testing the Validity of a ClaimTesting the Validity of a Claim

• Manufacturers’ claims are usually given the benefitManufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis.of the doubt and stated as the null hypothesis.

• The conclusion that the claim is false comes fromThe conclusion that the claim is false comes from sample data that contradict the null hypothesis.sample data that contradict the null hypothesis.

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Testing in Decision-Making SituationsTesting in Decision-Making Situations


• A decision maker might have to choose betweenA decision maker might have to choose between two courses of action, one associated with the nulltwo courses of action, one associated with the null hypothesis and another associated with thehypothesis and another associated with the alternative hypothesis.alternative hypothesis.

• Example: Accepting a shipment of goods from aExample: Accepting a shipment of goods from a supplier or returning the shipment of goods to thesupplier or returning the shipment of goods to the suppliersupplier

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One-tailedOne-tailed(lower-tail)(lower-tail)

One-tailedOne-tailed(upper-tail)(upper-tail)

Two-tailedTwo-tailed

0 0: H 0 0: H

0: aH 0: aH 0 0: H 0 0: H

0: aH 0: aH 0 0: H 0 0: H

0: aH 0: aH

Summary of Forms for Null and Summary of Forms for Null and Alternative Hypotheses about a Alternative Hypotheses about a

Population MeanPopulation Mean The equality part of the hypotheses always appearsThe equality part of the hypotheses always appears

in the null hypothesis.in the null hypothesis. In general, a hypothesis test about the value of aIn general, a hypothesis test about the value of a population mean population mean must take one of the followingmust take one of the following three forms (where three forms (where 00 is the hypothesized value of is the hypothesized value of the population mean).the population mean).

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Example: Metro EMSExample: Metro EMS

Null and Alternative HypothesesNull and Alternative Hypotheses

A major west coast city provides one of A major west coast city provides one of the mostthe mostcomprehensive emergency medical services comprehensive emergency medical services in thein theworld. Operating in a multiple hospital world. Operating in a multiple hospital systemsystemwith approximately 20 mobile medical units, with approximately 20 mobile medical units, thetheservice goal is to respond to medical service goal is to respond to medical emergenciesemergencieswith a mean time of 12 minutes or less.with a mean time of 12 minutes or less.

The director of medical services wants toThe director of medical services wants toformulate a hypothesis test that could use a formulate a hypothesis test that could use a samplesampleof emergency response times to determine of emergency response times to determine whetherwhetheror not the service goal of 12 minutes or less or not the service goal of 12 minutes or less is beingis beingachieved.achieved.

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Null and Alternative HypothesesNull and Alternative Hypotheses

The emergency service is meetingThe emergency service is meetingthe response goal; no follow-upthe response goal; no follow-upaction is necessary.action is necessary.

The emergency service is notThe emergency service is notmeeting the response goal;meeting the response goal;appropriate follow-up action isappropriate follow-up action isnecessary.necessary.

HH00: :

HHaa::

where: where: = mean response time for the population = mean response time for the population of medical emergency requestsof medical emergency requests

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Type I ErrorType I Error

Because hypothesis tests are based on sample data,Because hypothesis tests are based on sample data, we must allow for the possibility of errors.we must allow for the possibility of errors. A A Type I errorType I error is rejecting is rejecting HH00 when it is true. when it is true.

The probability of making a Type I error when theThe probability of making a Type I error when the

null hypothesis is true as an equality is called thenull hypothesis is true as an equality is called the

level of significancelevel of significance..

Applications of hypothesis testing that only controlApplications of hypothesis testing that only control

the Type I error are often called the Type I error are often called significance testssignificance tests..

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Type II ErrorType II Error

A A Type II errorType II error is accepting is accepting HH00 when it is false. when it is false.

It is difficult to control for the probability of makingIt is difficult to control for the probability of making

a Type II error.a Type II error.

Statisticians avoid the risk of making a Type IIStatisticians avoid the risk of making a Type II

error by using “do not reject error by using “do not reject HH00” and not “accept ” and not “accept HH00”.”.

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Type I and Type II ErrorsType I and Type II Errors

CorrectCorrectDecisionDecision Type II ErrorType II Error

CorrectCorrectDecisionDecisionType I ErrorType I Error

RejectReject HH00

(Conclude (Conclude > 12) > 12)

AcceptAccept HH00

(Conclude(Conclude << 12) 12)

HH0 0 TrueTrue(( << 12) 12)

HH0 0 FalseFalse

(( > 12) > 12)ConclusionConclusion

Population Condition Population Condition

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pp-Value Approach to-Value Approach toOne-Tailed Hypothesis TestingOne-Tailed Hypothesis Testing

Reject Reject HH00 if the if the pp-value -value << ..

The The pp-value-value is the probability, computed using the is the probability, computed using the test statistic, that measures the support (or lack oftest statistic, that measures the support (or lack of support) provided by the sample for the nullsupport) provided by the sample for the null hypothesis.hypothesis.

If the If the pp-value is less than or equal to the level of-value is less than or equal to the level of significance significance , the value of the test statistic is in the, the value of the test statistic is in the rejection region.rejection region.

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pp-Value Approach-Value Approach pp-Value Approach-Value Approach

p-valuep-value

00 -z = -1.28 -z = -1.28

= .10 = .10

zz

z =-1.46 z =-1.46

Lower-Tailed Test About a Population Lower-Tailed Test About a Population Mean:Mean:

KnownKnown

Samplingdistribution

of


of z xn

0

/z x

n

0

/

pp-Value -Value << ,,

so reject so reject HH00..

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pp-Value Approach-Value Approach pp-Value Approach-Value Approach

p-Valuep-Value

00 z = 1.75 z = 1.75

= .04 = .04

zz

z =2.29 z =2.29

Upper-Tailed Test About a Population Upper-Tailed Test About a Population Mean:Mean:

KnownKnown


of


of z xn

0

/z x

n

0

/

pp-Value -Value << ,,so reject so reject HH00..

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Critical Value Approach to Critical Value Approach to One-Tailed Hypothesis TestingOne-Tailed Hypothesis Testing

The test statistic The test statistic zz has a standard normal probability has a standard normal probability distribution.distribution.

We can use the standard normal probabilityWe can use the standard normal probability distribution table to find the distribution table to find the zz-value with an -value with an areaarea of of in the lower (or upper) tail of the in the lower (or upper) tail of the distribution.distribution.

The value of the test statistic that established theThe value of the test statistic that established the boundary of the rejection region is called theboundary of the rejection region is called the critical valuecritical value for the test. for the test.

The rejection rule is:The rejection rule is:• Lower tail: Reject Lower tail: Reject HH00 if if zz << - -zz

• Upper tail: Reject Upper tail: Reject HH00 if if zz >> zz

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00 z = 1.28 z = 1.28

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

zz


of


of z xn

0

/z x

n

0

/

Lower-Tailed Test About a Population Lower-Tailed Test About a Population Mean:Mean:

KnownKnown Critical Value ApproachCritical Value Approach Critical Value ApproachCritical Value Approach

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00 z = 1.645 z = 1.645

Reject H0Reject H0


zz


of


of z xn

0

/z x

n

0

/

Upper-Tailed Test About a Population Upper-Tailed Test About a Population Mean:Mean:

KnownKnown Critical Value ApproachCritical Value Approach Critical Value ApproachCritical Value Approach

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Steps of Hypothesis TestingSteps of Hypothesis Testing

Step 1.Step 1. Develop the null and alternative hypotheses. Develop the null and alternative hypotheses.

Step 2.Step 2. Specify the level of significance Specify the level of significance ..

Step 3.Step 3. Collect the sample data and compute Collect the sample data and compute the test statistic.the test statistic.

pp-Value Approach-Value Approach

Step 4.Step 4. Use the value of the test statistic to compute the Use the value of the test statistic to compute the pp-value.-value.

Step 5.Step 5. Reject Reject HH00 if if pp-value -value << ..

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Critical Value ApproachCritical Value Approach

Step 4.Step 4. Use the level of significance Use the level of significanceto to determine the critical value and the determine the critical value and the rejection rule.rejection rule.

Step 5.Step 5. Use the value of the test statistic and the Use the value of the test statistic and the rejectionrejection

rule to determine whether to reject rule to determine whether to reject HH00..

Steps of Hypothesis TestingSteps of Hypothesis Testing

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Example: Metro EMSExample: Metro EMS

The EMS director wants to perform a The EMS director wants to perform a hypothesishypothesistest, with a .05 level of significance, to test, with a .05 level of significance, to determinedeterminewhether the service goal of 12 minutes or less whether the service goal of 12 minutes or less is is being achieved.being achieved.

The response times for a random sample The response times for a random sample of 40of 40medical emergencies were tabulated. The medical emergencies were tabulated. The samplesamplemean is 13.25 minutes. The population mean is 13.25 minutes. The population standardstandarddeviation is believed to be 3.2 minutes.deviation is believed to be 3.2 minutes.

One-Tailed Tests About a Population One-Tailed Tests About a Population Mean:Mean:

KnownKnown

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1. Develop the hypotheses.1. Develop the hypotheses.

2. Specify the level of significance.2. Specify the level of significance. = .05= .05

HH00: :

HHaa::

pp -Value and Critical Value Approaches -Value and Critical Value Approaches


KnownKnown

3. Compute the value of the test statistic.3. Compute the value of the test statistic.

13.25 12 2.47

/ 3.2/ 40x

zn

13.25 12 2.47

/ 3.2/ 40x

zn

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5. Determine whether to reject 5. Determine whether to reject HH00..

pp –Value Approach –Value Approach


KnownKnown

4. Compute the 4. Compute the pp –value. –value.

For For zz = 2.47, cumulative probability = .9932. = 2.47, cumulative probability = .9932.

pp–value = 1 –value = 1 .9932 = .0068 .9932 = .0068

Because Because pp–value = .0068 –value = .0068 << = .05, we reject = .05, we reject HH00..

There is sufficient statistical There is sufficient statistical evidenceevidence

to infer that Metro EMS is to infer that Metro EMS is notnot meetingmeeting

the response goal of 12 minutes.the response goal of 12 minutes.

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p p –Value Approach–Value Approach p p –Value Approach–Value Approach

p-valuep-value

00 z =1.645 z =1.645

= .05 = .05

zz

z =2.47 z =2.47


KnownKnown


of


of z xn

0

/z x

n

0

/

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Excel Formula Excel Formula WorksheetWorksheet


KnownKnown

A B C

1Response

Time Sample Size =COUNT(A2:A41)2 19.5 Sample Mean =AVERAGE(A2:A41)3 15.24 11.0 Population Std. Dev. 3.25 12.8 Hypothesized Value 126 12.47 20.3 Standard Error =C4/SQRT(C1)8 9.6 Test Statistic z =(C2-C5)/C79 10.910 16.2 p -Value (lower tail) =NORMSDIST(C8)11 13.4 p -Value (lower tail) =1-C1012 19.7 p -Value (two tail) =2*(MIN(C10,C11))

Note:Note:Rows 13-41Rows 13-41

are not shown.are not shown.

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Excel Value Excel Value WorksheetWorksheet


KnownKnown

A B C

1Response

Time Sample Size 402 19.5 Sample Mean 13.253 15.24 11.0 Population Std. Dev. 3.25 12.8 Hypothesized Value 126 12.47 20.3 Standard Error 0.5068 9.6 Test Statistic z 2.479 10.910 16.2 p -Value (lower tail) 0.993311 13.4 p -Value (upper tail) 0.006712 19.7 p -Value (two tail) 0.0134



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There is sufficient statistical There is sufficient statistical evidenceevidence

to infer that Metro EMS is to infer that Metro EMS is notnot meetingmeeting

the response goal of 12 minutes.the response goal of 12 minutes.

Because 2.47 Because 2.47 >> 1.645, we reject 1.645, we reject HH00..



KnownKnown

For For = .05, = .05, zz.05.05 = 1.645 = 1.645

4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.

Reject Reject HH00 if if zz >> 1.645 1.645

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pp-Value Approach to-Value Approach toTwo-Tailed Hypothesis TestingTwo-Tailed Hypothesis Testing

The rejection rule:The rejection rule: Reject Reject HH00 if the if the pp-value -value << ..

Compute the Compute the pp-value-value using the following three steps: using the following three steps:

3. Double the tail area obtained in step 2 to obtain3. Double the tail area obtained in step 2 to obtain the the pp –value. –value.

2. If 2. If zz is in the upper tail ( is in the upper tail (zz > 0), find the area under > 0), find the area under the standard normal curve to the right of the standard normal curve to the right of zz..

If If zz is in the lower tail ( is in the lower tail (zz < 0), find the area under < 0), find the area under the standard normal curve to the left of the standard normal curve to the left of zz..

1. Compute the value of the test statistic 1. Compute the value of the test statistic zz..

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Critical Value Approach to Critical Value Approach to Two-Tailed Hypothesis TestingTwo-Tailed Hypothesis Testing

The critical values will occur in both the lower andThe critical values will occur in both the lower and upper tails of the standard normal curve.upper tails of the standard normal curve.

The rejection rule is:The rejection rule is:

Reject Reject HH00 if if zz << - -zz/2/2 or or zz >> zz/2/2..

Use the standard normal probability Use the standard normal probability distributiondistribution table to find table to find zz/2/2 (the (the zz-value with an area of -value with an area of /2 in/2 in the upper tail of the distribution).the upper tail of the distribution).

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Example: Glow ToothpasteExample: Glow Toothpaste

Quality assurance procedures call for theQuality assurance procedures call for thecontinuation of the filling process if the continuation of the filling process if the samplesampleresults are consistent with the assumption results are consistent with the assumption that thethat themean filling weight for the population of mean filling weight for the population of toothpastetoothpastetubes is 6 oz.; otherwise the process will be tubes is 6 oz.; otherwise the process will be adjusted.adjusted.

The production line for Glow toothpaste isThe production line for Glow toothpaste isdesigned to fill tubes with a mean weight of 6 designed to fill tubes with a mean weight of 6 oz.oz.Periodically, a sample of 30 tubes will be Periodically, a sample of 30 tubes will be selected inselected inorder to check the filling process.order to check the filling process.

Two-Tailed Tests About a Population Two-Tailed Tests About a Population Mean:Mean:

KnownKnown

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Perform a hypothesis test, at the .03 level Perform a hypothesis test, at the .03 level ofofsignificance, to help determine whether the significance, to help determine whether the fillingfillingprocess should continue operating or be process should continue operating or be stopped andstopped andcorrected.corrected.

Assume that a sample of 30 toothpaste Assume that a sample of 30 toothpaste tubestubesprovides a sample mean of 6.1 oz. The provides a sample mean of 6.1 oz. The populationpopulationstandard deviation is believed to be 0.2 oz.standard deviation is believed to be 0.2 oz.


KnownKnown Example: Glow ToothpasteExample: Glow Toothpaste

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1. Determine the hypotheses.1. Determine the hypotheses.

2. Specify the level of significance.2. Specify the level of significance.


= .03= .03

pp –Value and Critical Value Approaches –Value and Critical Value Approaches

HH00: :

HHaa:: 6 6


KnownKnown

0 6.1 6

2.74/ .2/ 30

xz

n

0 6.1 6

2.74/ .2/ 30

xz

n

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KnownKnown




For For zz = 2.74, cumulative probability = .9969 = 2.74, cumulative probability = .9969

pp–value = 2(1 –value = 2(1 .9969) = .0062 .9969) = .0062

Because Because pp–value = .0062 –value = .0062 << = .03, we reject = .03, we reject HH00..

There is sufficient statistical evidence toThere is sufficient statistical evidence toinfer that the alternative hypothesis is infer that the alternative hypothesis is

truetrue (i.e. the mean filling weight is not 6 (i.e. the mean filling weight is not 6

ounces).ounces).

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KnownKnown

/2 = .015/2 = .015

00z/2 = 2.17z/2 = 2.17

zz

/2 = .015/2 = .015

pp-Value Approach-Value Approach

-z/2 = -2.17-z/2 = -2.17z = 2.74z = 2.74z = -2.74z = -2.74

1/2p -value= .0031

1/2p -value= .0031

1/2p -value= .0031

1/2p -value= .0031

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Excel Formula WorksheetExcel Formula Worksheet

Note: Rows 13-31 are not shown.Note: Rows 13-31 are not shown.


KnownKnown

A B C1 Weight Sample Size =COUNT(A2:A31)2 6.04 Sample Mean =AVERAGE(A2:A31)3 5.994 5.92 Population Std. Dev. 0.25 6.03 Hypothesized Value 66 6.017 5.95 Standard Error =C4/SQRT(C1)8 6.09 Test Statistic z =(C2-C5)/C79 6.0710 6.07 p -Value (lower tail) =NORMSDIST(C8)11 5.97 p -Value (upper tail) =1-C1012 5.96 p -Value (two tail) =2*(MIN(C10,C11))

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Excel Value WorksheetExcel Value Worksheet


KnownKnown

A B C1 Weight Sample Size 302 6.04 Sample Mean 6.13 5.994 5.92 Population Std. Dev. 0.25 6.03 Hypothesized Value 66 6.017 5.95 Standard Error 0.03658 6.09 Test Statistic z 2.73869 6.0710 6.07 p -Value (lower tail) 0.996911 5.97 p -Value (upper tail) 0.003112 5.96 p -Value (two tail) 0.0062


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KnownKnown


There is sufficient statistical evidence toThere is sufficient statistical evidence toinfer that the alternative hypothesis is infer that the alternative hypothesis is

truetrue (i.e. the mean filling weight is not 6 (i.e. the mean filling weight is not 6

ounces).ounces).


For For /2 = .03/2 = .015, /2 = .03/2 = .015, zz.015.015 = 2.17 = 2.17


Reject Reject HH00 if if zz << -2.17 or -2.17 or zz >> 2.17 2.17

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/2 = .015/2 = .015

00 2.17 2.17

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

-2.17 -2.17



of


of z xn

0

/z x

n

0

/


KnownKnown

/2 = .015/2 = .015

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Confidence Interval Approach toConfidence Interval Approach toTwo-Tailed Tests About a Population MeanTwo-Tailed Tests About a Population Mean Select a simple random sample from the populationSelect a simple random sample from the population and use the value of the sample mean to developand use the value of the sample mean to develop the confidence interval for the population mean the confidence interval for the population mean .. (Confidence intervals are covered in Chapter 8.)(Confidence intervals are covered in Chapter 8.)

xx

If the confidence interval contains the hypothesizedIf the confidence interval contains the hypothesized value value 00, do not reject , do not reject HH00. Otherwise, reject . Otherwise, reject HH00..

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The 97% confidence interval for The 97% confidence interval for is is

/ 2 6.1 2.17(.2 30) 6.1 .07924x zn

/ 2 6.1 2.17(.2 30) 6.1 .07924x zn

Confidence Interval Approach toConfidence Interval Approach toTwo-Tailed Tests About a Population MeanTwo-Tailed Tests About a Population Mean

Because the hypothesized value for theBecause the hypothesized value for the

population mean, population mean, 00 = 6, is not in this interval, = 6, is not in this interval,the hypothesis-testing conclusion is that thethe hypothesis-testing conclusion is that the

null hypothesis, null hypothesis, HH00: : = 6, can be rejected. = 6, can be rejected.

or 6.02076 to 6.17924or 6.02076 to 6.17924

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Test StatisticTest Statistic

Tests About a Population Mean:Tests About a Population Mean: Unknown Unknown

txs n

0/

txs n

0/

This test statistic has a This test statistic has a tt distribution distribution with with nn - 1 degrees of freedom. - 1 degrees of freedom.

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Rejection Rule: Rejection Rule: pp -Value Approach -Value Approach

HH00: : Reject Reject HH0 0 if if tt >> tt

Reject Reject HH0 0 if if tt << - -tt

Reject Reject HH0 0 if if tt << - - tt or or tt >> tt

HH00: :

HH00: :

Tests About a Population Mean:Tests About a Population Mean: Unknown Unknown

Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach

Reject Reject HH0 0 if if p p –value –value <<

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p p -Values and the -Values and the tt Distribution Distribution

The format of the The format of the tt distribution table provided in most distribution table provided in most statistics textbooks does not have sufficient detailstatistics textbooks does not have sufficient detail to determine the to determine the exactexact p p-value for a hypothesis test.-value for a hypothesis test.

However, we can still use the However, we can still use the tt distribution table to distribution table to identify a identify a rangerange for the for the pp-value.-value.

An advantage of computer software packages is thatAn advantage of computer software packages is that the computer output will provide the the computer output will provide the pp-value for the-value for the tt distribution. distribution.

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A State Highway Patrol periodically A State Highway Patrol periodically samplessamples

vehicle speeds at various locations on a vehicle speeds at various locations on a particularparticular

roadway. The sample of vehicle speeds is roadway. The sample of vehicle speeds is used toused to

test the hypothesis test the hypothesis HH00: : << 65. 65.

Example: Highway PatrolExample: Highway Patrol

One-Tailed Test About a Population Mean: One-Tailed Test About a Population Mean: Unknown Unknown

The locations where The locations where HH00 is rejected are is rejected are deemed thedeemed the

best locations for radar traps. At Location F, abest locations for radar traps. At Location F, a

sample of 64 vehicles shows a mean speed of sample of 64 vehicles shows a mean speed of 66.266.2

mph with a standard deviation of 4.2 mph. mph with a standard deviation of 4.2 mph. Use Use = .05 to test the hypothesis.= .05 to test the hypothesis.

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One-Tailed Test About a Population Mean:One-Tailed Test About a Population Mean: Unknown Unknown




= .05= .05


HH00: : << 65 65

HHaa: : > 65 > 65

0 66.2 65

2.286/ 4.2/ 64

xt

s n

0 66.2 65

2.286/ 4.2/ 64

xt

s n

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For For tt = 2.286, the = 2.286, the pp–value must be less than .025–value must be less than .025(for (for tt = 1.998) and greater than .01 (for = 1.998) and greater than .01 (for tt = 2.387). = 2.387).

.01 < .01 < pp–value < .025–value < .025

Because Because pp–value –value << = .05, we reject = .05, we reject HH00..

We are at least 95% confident that the mean We are at least 95% confident that the mean speedspeed of vehicles at Location F is greater than 65 of vehicles at Location F is greater than 65 mph.mph.

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A B C1 Speed Sample Size =COUNT(A2:A65)2 69.6 Sample Mean =AVERAGE(A2:A65)3 73.5 Sample Std. Dev. =STDEV(A2:A65)4 74.1 5 64.4 Hypoth. Value 656 66.37 68.7 Standard Error =C3/SQRT(C1)8 69.0 Test Statistic t =(C2-C5)/C79 65.2 Degr. of Freedom =C1-110 71.111 70.8 p -Value (lower tail) =IF(C8<0,TDIST(-C8,C9,1),1-TDIST(C8,C9,1))12 64.6 p -Value (upper tail) =1-C1113 67.4 p -Value (two tail) =2*MIN(C11,C12)

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Excel Value WorksheetExcel Value Worksheet



A B C1 Speed Sample Size 642 69.6 Sample Mean 66.23 73.5 Sample Std. Dev. 4.24 74.1 5 64.4 Hypoth. Value 656 66.37 68.7 Standard Error 0.5258 69.0 Test Statistic t 2.2869 65.2 Degr. of Freedom 6310 71.111 70.8 p -Value (lower tail) 0.987212 64.6 p -Value (upper tail) 0.012813 67.4 p -Value (two tail) 0.0256

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We are at least 95% confident that the mean We are at least 95% confident that the mean speed of vehicles at Location F is greater speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate than 65 mph. Location F is a good candidate for a radar trap.for a radar trap.



For For = .05 and d.f. = 64 – 1 = 63, = .05 and d.f. = 64 – 1 = 63, tt.05.05 = 1.669 = 1.669


Reject Reject HH00 if if tt >> 1.669 1.669

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00 t =1.669 t =1.669

Reject H0Reject H0


tt


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The equality part of the hypotheses always appearsThe equality part of the hypotheses always appears in the null hypothesis.in the null hypothesis. In general, a hypothesis test about the value of aIn general, a hypothesis test about the value of a population proportion population proportion pp must take one of themust take one of the

following three forms (where following three forms (where pp00 is the hypothesized is the hypothesized

value of the population proportion).value of the population proportion).

A Summary of Forms for Null and A Summary of Forms for Null and Alternative Hypotheses About a Alternative Hypotheses About a

Population ProportionPopulation Proportion

One-tailedOne-tailed(lower tail)(lower tail)

One-tailedOne-tailed(upper tail)(upper tail)

Two-tailedTwo-tailed

0 0: H p p0 0: H p p

0: aH p p 0: aH p p0: aH p p 0: aH p p0 0: H p p0 0: H p p 0 0: H p p0 0: H p p

0: aH p p 0: aH p p

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Test StatisticTest Statistic

zp p

p

0

z

p p

p

0

pp p

n 0 01( ) pp p

n 0 01( )

Tests About a Population ProportionTests About a Population Proportion

where:where:

assuming assuming npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5 5

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Rejection Rule: Rejection Rule: pp –Value Approach –Value Approach

HH00: : pppp Reject Reject HH0 0 if if zz >> zz

Reject Reject HH0 0 if if zz << - -zz

Reject Reject HH0 0 if if zz << - -zz or or zz >> zz

HH00: : pppp

HH00: : pppp

Tests About a Population ProportionTests About a Population Proportion

Reject Reject HH0 0 if if p p –value –value <<

Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach

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Example: National Safety Council Example: National Safety Council (NSC)(NSC) For a Christmas and New Year’s week, For a Christmas and New Year’s week, thethe

National Safety Council estimated that 500 National Safety Council estimated that 500 peoplepeople

would be killed and 25,000 injured on the would be killed and 25,000 injured on the nation’snation’s

roads. The NSC claimed that 50% of the roads. The NSC claimed that 50% of the accidentsaccidents

would be caused by drunk driving.would be caused by drunk driving.

Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion

A sample of 120 accidents showed that A sample of 120 accidents showed that 67 were67 were

caused by drunk driving. Use these data to caused by drunk driving. Use these data to test thetest the

NSC’s claim with NSC’s claim with = .05. = .05.

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= .05= .05


0: .5H p0: .5H p: .5aH p: .5aH p

0 (67/ 120) .5 1.28

.045644p

p pz

0 (67/ 120) .5 1.28

.045644p

p pz

0 0(1 ) .5(1 .5).045644

120p

p p

n

0 0(1 ) .5(1 .5)

.045644120p

p p

n

a commona commonerror is error is usingusing

in this in this formula formula

a commona commonerror is error is usingusing

in this in this formula formula

pp

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ppValue ApproachValue Approach

4. Compute the 4. Compute the pp -value. -value.


Because Because pp–value = .2006 > –value = .2006 > = .05, we cannot reject = .05, we cannot reject HH00..


For For zz = 1.28, cumulative probability = .8997 = 1.28, cumulative probability = .8997

pp–value = 2(1 –value = 2(1 .8997) = .2006 .8997) = .2006

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Using Excel to Conduct HypothesisUsing Excel to Conduct HypothesisTests about a Population ProportionTests about a Population Proportion

A B C

1Drunk

Driving Sample Size =COUNT(A2:A121)2 No Response of Interest Yes3 Yes Count for Response =COUNTIF(A2:A121,C2)4 No Sample Proportion =C3/C15 Yes6 No Hypothesized Value 0.57 Yes8 Yes Standard Error =SQRT(C6*(1-C6)/C1)9 No Test Statistic z =(C4-C6)/C810 No11 Yes p -Value (lower tail) =NORMSDIST(C9)12 Yes p -Value (upper tail) =1-C1113 Yes p -Value (two tail) =2*MIN(C11,C12)



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Using Excel to Conduct HypothesisUsing Excel to Conduct HypothesisTests about a Population ProportionTests about a Population Proportion

Excel Value WorksheetExcel Value WorksheetA B C

1Drunk

Driving Sample Size 1202 No Response of Interest Yes3 Yes Count for Response 674 No Sample Proportion 0.55835 Yes6 No Hypothesized Value 0.57 Yes8 Yes Standard Error 0.04569 No Test Statistic z 1.2810 No11 Yes p -Value (lower tail) 0.899712 Yes p -Value (upper tail) 0.100313 Yes p -Value (two tail) 0.2006



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For For /2 = .05/2 = .025, /2 = .05/2 = .025, zz.025.025 = 1.96 = 1.96

4. Determine the criticals value and 4. Determine the criticals value and rejection rule.rejection rule.

Reject Reject HH00 if if zz << -1.96 or -1.96 or zz >> 1.96 1.96

Because 1.278 > -1.96 and < 1.96, we cannot reject Because 1.278 > -1.96 and < 1.96, we cannot reject HH00..

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End of Chapter 9End of Chapter 9

Slides by JOHN LOUCKS St. Edward’s University

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