Slides Prepared by JOHN S. LOUCKS - Cameron Universitycameron.edu/~syeda/orgl333/ch9.pdf · 2007-08-17 · © 2006 Thomson/South-Western SlideSlide 11 Slides Prepared by JOHN S. LOUCKS

11SlideSlide©© 2006 Thomson/South2006 Thomson/South--WesternWestern

Slides Prepared by

JOHN S. LOUCKSSt. Edward’s University

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. EdwardSt. Edward’’s Universitys University


Chapter 9Chapter 9Hypothesis TestingHypothesis Testing

Developing Null and Alternative HypothesesDeveloping Null and Alternative HypothesesType I and Type II ErrorsType I and Type II ErrorsPopulation Mean: Population Mean: σσ KnownKnownPopulation Mean: Population Mean: σσ UnknownUnknownPopulation ProportionPopulation Proportion


Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses

Hypothesis testingHypothesis testing can be used to determine whethercan be used to determine whethera statement about the value of a population parametera statement about the value of a population parametershould 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 tentativeassumption about a population parameter.assumption about a population parameter.The The alternative hypothesisalternative hypothesis, denoted by , denoted by HHaa, is the, is theopposite 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 isattempting to establish.attempting to establish.


Testing Research HypothesesTesting Research Hypotheses


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

•• The conclusion that the research hypothesis is trueThe conclusion that the research hypothesis is truecomes from sample data that contradict the nullcomes from sample data that contradict the nullhypothesis.hypothesis.



Testing the Validity of a ClaimTesting the Validity of a Claim

•• ManufacturersManufacturers’’ claims are usually given the benefitclaims are usually given the benefitof 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 fromsample data that contradict the null hypothesis.sample data that contradict the null hypothesis.


Testing in DecisionTesting in Decision--Making SituationsMaking Situations


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

•• Example: Accepting a shipment of goods from aExample: Accepting a shipment of goods from asupplier or returning the shipment of goods to thesupplier or returning the shipment of goods to thesuppliersupplier


OneOne--tailedtailed(lower(lower--tail)tail)

OneOne--tailedtailed(upper(upper--tail)tail)

TwoTwo--tailedtailed

0 0: H μ μ≥0 0: H μ μ≥

0: aH μ μ< 0: aH μ μ<0 0: H μ μ≤0 0: H μ μ≤

0: aH μ μ> 0: aH μ μ>0 0: H μ μ=0 0: H μ μ=

0: aH μ μ≠ 0: aH μ μ≠

Summary of Forms for Null and Alternative Summary of Forms for Null and Alternative Hypotheses about a Population MeanHypotheses about a Population Mean

The equality part of the hypotheses always appearsThe equality part of the hypotheses always appearsin 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 apopulation mean population mean μμ must take one of the followingmust take one of the followingthree forms (where three forms (where μμ00 is the hypothesized value ofis the hypothesized value ofthe population mean).the population mean).


Example: Metro EMSExample: Metro EMS

Null and Alternative HypothesesNull and Alternative Hypotheses

Operating in a multipleOperating in a multiplehospital system with hospital system with approximately 20 mobile medicalapproximately 20 mobile medicalunits, the service goal is to respond to medicalunits, the service goal is to respond to medicalemergencies with a mean time of 12 minutes or less.emergencies with a mean time of 12 minutes or less.

A major west coast city providesA major west coast city providesone of the most comprehensiveone of the most comprehensiveemergency medical services inemergency medical services inthe world.the world.


The director of medical servicesThe director of medical serviceswants to formulate a hypothesiswants to formulate a hypothesistest that could use a sample oftest that could use a sample ofemergency response times toemergency response times todetermine whether or not thedetermine whether or not theservice goal of 12 minutes or lessservice goal of 12 minutes or lessis being achieved.is being achieved.





The emergency service is meetingThe emergency service is meetingthe response goal; no followthe response goal; no follow--upupaction is necessary.action is necessary.

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

HH00: : μμ << 12 12

HHaa:: μμ > 12 > 12

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


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 thenull hypothesis is true as an equality is called thenull hypothesis is true as an equality is called thelevel of significancelevel of significance..

Applications of hypothesis testing that only controlApplications of hypothesis testing that only controlthe Type I error are often called the Type I error are often called significance testssignificance tests..


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 makinga Type II error.a Type II error.

Statisticians avoid the risk of making a Type IIStatisticians avoid the risk of making a Type IIerror by using error by using ““do not reject do not reject HH00”” and not and not ““accept accept HH00””..


Type I and Type II ErrorsType I and Type II Errors

CorrectCorrectDecisionDecision Type II ErrorType II Error

CorrectCorrectDecisionDecisionType I ErrorType I ErrorRejectReject HH00

(Conclude (Conclude μμ > 12)> 12)

AcceptAccept HH00(Conclude(Conclude μμ << 12)12)

HH0 0 TrueTrue((μμ << 12)12)

HH0 0 FalseFalse((μμ > 12)> 12)ConclusionConclusion

Population Condition Population Condition


pp--Value Approach toValue Approach toOneOne--Tailed Hypothesis TestingTailed Hypothesis Testing

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

The The pp--valuevalue is the probability, computed using theis the probability, computed using thetest statistic, that measures the support (or lack oftest statistic, that measures the support (or lack ofsupport) provided by the sample for the nullsupport) provided by the sample for the nullhypothesis.hypothesis.

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


p-Value Approachpp--Value ApproachValue Approach

p-value= .072

p-value= .072

00-zα =-1.28-zα =-1.28

α = .10α = .10

zz

z =-1.46z =-1.46

LowerLower--Tailed Test About a Population Mean:Tailed Test About a Population Mean:σσ KnownKnown

Samplingdistributionof

Samplingdistributionof z x

n= − μσ

0/

z xn= − μ

σ0

/

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


p-Value Approachpp--Value ApproachValue Approach

p-Value= .011

p-Value= .011

00 zα =1.75zα =1.75

α = .04α = .04

zzz =2.29z =2.29

UpperUpper--Tailed Test About a Population Mean:Tailed Test About a Population Mean:σσ KnownKnown



n= − μσ

0/

z xn= − μ

σ0

/

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


Critical Value Approach to Critical Value Approach to OneOne--Tailed Hypothesis TestingTailed Hypothesis Testing

The test statistic The test statistic zz has a standard normal probabilityhas a standard normal probabilitydistribution.distribution.We can use the standard normal probabilityWe can use the standard normal probabilitydistribution table to find the distribution table to find the zz--value with an areavalue with an areaof of αα in the lower (or upper) tail of the distribution.in the lower (or upper) tail of the distribution.The value of the test statistic that established theThe value of the test statistic that established theboundary of the rejection region is called theboundary of the rejection region is called thecritical 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 << --zzαα

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


α = .10α = .10

00−zα = −1.28−zα = −1.28

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

zz



n= − μσ

0/

z xn= − μ

σ0

/

LowerLower--Tailed Test About a Population Mean:Tailed Test About a Population Mean:σσ KnownKnown

Critical Value ApproachCritical Value ApproachCritical Value Approach


α = .05α = .05

00 zα = 1.645zα = 1.645

Reject H0Reject H0


zz



n= − μσ

0/

z xn= − μ

σ0

/

UpperUpper--Tailed Test About a Population Mean:Tailed Test About a Population Mean:σσ KnownKnown

Critical Value ApproachCritical Value ApproachCritical Value Approach


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

statistic.statistic.

pp--Value ApproachValue Approach

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

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


Critical Value ApproachCritical Value ApproachStep 4.Step 4. Use the level of significanceUse the level of significance to determine the to determine the

critical value and the rejection rule.critical value and the rejection rule.

Step 5.Step 5. Use the value of the test statistic and the rejectionUse the value of the test statistic and the rejectionrule to determine whether to reject rule to determine whether to reject HH00..

Steps of Hypothesis TestingSteps of Hypothesis Testing



The EMS director wants toThe EMS director wants toperform a hypothesis test, with aperform a hypothesis test, with a.05 level of significance, to determine.05 level of significance, to determinewhether the service goal of 12 minutes or less is beingwhether the service goal of 12 minutes or less is beingachieved.achieved.

The response times for a randomThe response times for a randomsample of 40 medical emergenciessample of 40 medical emergencieswere tabulated. The sample meanwere tabulated. The sample meanis 13.25 minutes. The populationis 13.25 minutes. The populationstandard deviation is believed tostandard deviation is believed tobe 3.2 minutes.be 3.2 minutes.

OneOne--Tailed Tests About a Population Mean:Tailed Tests About a Population Mean:σσ KnownKnown


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

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

HH00: : μμ << 12 12HHaa:: μμ > 12 > 12

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


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

μσ

− −= = =

13.25 12 2.47/ 3.2/ 40

xzn

μσ

− −= = =

13.25 12 2.47/ 3.2/ 40

xzn


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

We are at least 95% confident that Metro EMS We are at least 95% confident that Metro EMS is is notnot meeting the response goal of 12 minutes.meeting the response goal of 12 minutes.

pp ––Value ApproachValue Approach


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 << αα = .05, we reject = .05, we reject HH00..


p –Value Approachp p ––Value ApproachValue Approach

p-value= .0068p-value= .0068

00 zα =1.645zα =

1.645

α = .05α = .05

zz

z =2.47z =2.47




n= − μσ

0/

z xn= − μ

σ0

/



We are at least 95% confident that Metro EMS We are at least 95% confident that Metro EMS is is notnot meeting the response goal of 12 minutes.meeting the response goal of 12 minutes.

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

Critical Value ApproachCritical Value Approach


For For αα = .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 >> 1.6451.645


pp--Value Approach toValue Approach toTwoTwo--Tailed Hypothesis TestingTailed Hypothesis Testing

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

Compute the Compute the pp--valuevalue 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 obtainthe 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 underthe 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 underthe 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..


Critical Value Approach to Critical Value Approach to TwoTwo--Tailed Hypothesis TestingTailed Hypothesis Testing

The critical values will occur in both the lower andThe critical values will occur in both the lower andupper 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 << --zzαα/2/2 or or zz >> zzαα/2/2..

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


Example: Glow ToothpasteExample: Glow Toothpaste

TwoTwo--Tailed Test About a Population Mean: Tailed Test About a Population Mean: σσ KnownKnown

oz.

GlowGlow

Quality assurance procedures call forQuality assurance procedures call forthe continuation of the filling process if thethe continuation of the filling process if thesample results are consistent with the assumption thatsample results are consistent with the assumption thatthe mean filling weight for the population of toothpastethe mean filling weight for the population of toothpastetubes is 6 oz.; otherwise the process will be adjusted.tubes is 6 oz.; otherwise the process will be adjusted.

The production line for Glow toothpasteThe production line for Glow toothpasteis designed to fill tubes with a mean weightis designed to fill tubes with a mean weightof 6 oz. Periodically, a sample of 30 tubesof 6 oz. Periodically, a sample of 30 tubeswill be selected in order to check thewill be selected in order to check thefilling process.filling process.


Example: Glow ToothpasteExample: Glow Toothpaste

TwoTwo--Tailed Test About a Population Mean: Tailed Test About a Population Mean: σσ KnownKnown

oz.

GlowGlowPerform a hypothesis test, at the .03Perform a hypothesis test, at the .03level of significance, to help determinelevel of significance, to help determinewhether the filling process should continuewhether the filling process should continueoperating or be stopped and corrected.operating or be stopped and corrected.

Assume that a sample of 30 toothpasteAssume that a sample of 30 toothpastetubes provides a sample mean of 6.1 oz.tubes provides a sample mean of 6.1 oz.The population standard deviation is The population standard deviation is believed to be 0.2 oz.believed to be 0.2 oz.


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

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


α α = .03= .03

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

GlowGlow

HH00: : μμ = 6= 6HHaa:: 6μ ≠6μ ≠

TwoTwo--Tailed Tests About a Population Mean:Tailed Tests About a Population Mean:σσ KnownKnown

μσ

− −= = =0 6.1 6 2.74

/ .2 / 30xz

nμ

σ− −

= = =0 6.1 6 2.74/ .2 / 30

xzn


GlowGlow





For For zz = 2.74, cumulative probability = .9969= 2.74, cumulative probability = .9969pp––value = 2(1 value = 2(1 −− .9969) = .0062.9969) = .0062

Because Because pp––value = .0062 value = .0062 << αα = .03, we reject = .03, we reject HH00..We are at least 97% confident that the mean We are at least 97% confident that the mean

filling weight of the toothpaste tubes is not 6 oz.filling weight of the toothpaste tubes is not 6 oz.


GlowGlow


α/2 =.015

α/2 =.015

00zα/2 = 2.17zα/2 = 2.17

zz

α/2 =.015

α/2 =.015

pp--Value ApproachValue Approach

-zα/2 = -2.17-zα/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



GlowGlow



We are at least 97% confident that the mean We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz.filling weight of the toothpaste tubes is not 6 oz.


For For αα/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 >> 2.172.17


α/2 = .015α/2 = .015

00 2.172.17

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

-2.17-2.17

GlowGlow




n= − μσ

0/

z xn= − μ

σ0

/


α/2 = .015α/2 = .015


Confidence Interval Approach toConfidence Interval Approach toTwoTwo--Tailed Tests About a Population MeanTailed Tests About a Population MeanSelect a simple random sample from the populationSelect a simple random sample from the populationand use the value of the sample mean to developand use the value of the sample mean to developthe confidence interval for the population mean the confidence interval for the population mean μμ..(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 hypothesizedvalue value μμ00, do not reject , do not reject HH00. Otherwise, reject . Otherwise, reject HH00..


The 97% confidence interval for The 97% confidence interval for μμ isis

/2 6.1 2.17(.2 30) 6.1 .07924x znα

σ± = ± = ±/2 6.1 2.17(.2 30) 6.1 .07924x z

nασ

± = ± = ±

Confidence Interval Approach toConfidence Interval Approach toTwoTwo--Tailed Tests About a Population MeanTailed Tests About a Population Mean

GlowGlow

Because the hypothesized value for theBecause the hypothesized value for thepopulation mean, population mean, μμ00 = 6, is not in this interval,= 6, is not in this interval,the hypothesisthe hypothesis--testing conclusion is that thetesting conclusion is that thenull hypothesis, null hypothesis, HH00: : μμ = 6, can be rejected.= 6, can be rejected.

or 6.02076 to 6.17924or 6.02076 to 6.17924


Test StatisticTest Statistic

Tests About a Population Mean:Tests About a Population Mean:σσ UnknownUnknown

t xs n

=− μ0

/t x

s n=

− μ0

/

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


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

HH00: : μμ << μμ00 Reject Reject HH0 0 if if tt >> ttαα

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

Reject Reject HH0 0 if if tt << -- ttα/2α/2 or or tt >> ttα/2α/2

HH00: : μμ >> μμ00

HH00: : μμ = = μμ00

Tests About a Population Mean:Tests About a Population Mean:σσ UnknownUnknown

Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach

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


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

The format of the The format of the tt distribution table provided in mostdistribution table provided in moststatistics textbooks does not have sufficient detailstatistics textbooks does not have sufficient detailto determine the to determine the exactexact pp--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 todistribution table toidentify 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 thatthe computer output will provide the the computer output will provide the pp--value for thevalue for thett distribution.distribution.


A State Highway Patrol periodically samplesA State Highway Patrol periodically samplesvehicle speeds at various locationsvehicle speeds at various locationson a particular roadway. on a particular roadway. The sample of vehicle speedsThe sample of vehicle speedsis used to test the hypothesisis used to test the hypothesis

Example: Highway PatrolExample: Highway Patrol

OneOne--Tailed Test About a Population Mean: Tailed Test About a Population Mean: σσ UnknownUnknown

The locations where The locations where HH00 is rejected are deemedis rejected are deemedthe best locations for radar traps.the best locations for radar traps.

HH00: : μμ << 6565


Example: Highway PatrolExample: Highway Patrol

OneOne--Tailed Test About a Population Mean: Tailed Test About a Population Mean: σσ UnknownUnknownAt Location F, a sample of 64 vehicles shows aAt Location F, a sample of 64 vehicles shows a

mean speed of 66.2 mph with amean speed of 66.2 mph with astandard deviation ofstandard deviation of4.2 mph. Use 4.2 mph. Use αα = .05 to= .05 totest the hypothesis.test the hypothesis.


OneOne--Tailed Test About a Population Mean:Tailed Test About a Population Mean:σσ UnknownUnknown




α α = .05= .05


HH00: : μμ << 6565HHaa: : μμ > 65> 65

μ− −= = =0 66.2 65 2.286

/ 4.2 / 64xts n

μ− −= = =0 66.2 65 2.286

/ 4.2 / 64xts n






For For tt = 2.286, the = 2.286, the pp––value must be less than .025value 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 < .025value < .025

Because Because pp––value value << αα = .05, we reject = .05, we reject HH00..We are at least 95% confident that the mean speedWe are at least 95% confident that the mean speedof vehicles at Location F is greater than 65 mph.of vehicles at Location F is greater than 65 mph.




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



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


Reject Reject HH00 if if tt >> 1.6691.669


α = .05α = .05

00 tα =1.669tα =

1.669

Reject H0Reject H0


tt



The equality part of the hypotheses always appearsThe equality part of the hypotheses always appearsin 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 apopulation proportion population proportion pp must take one of themust take one of thefollowing three forms (where following three forms (where pp00 is the hypothesizedis the hypothesizedvalue of the population proportion).value of the population proportion).

A Summary of Forms for Null and Alternative A Summary of Forms for Null and Alternative Hypotheses About a Population ProportionHypotheses About a Population Proportion

OneOne--tailedtailed(lower tail)(lower tail)

OneOne--tailedtailed(upper tail)(upper tail)

TwoTwo--tailedtailed

0 0: H p p≥0 0: H p p≥

0: aH p p 0: aH p p> 0: aH p p>< 0: aH p p<0 0: H p p≤0 0: H p p≤ 0 0: H p p=0 0: H p p=

0: aH p p≠ 0: aH p p≠


Test StatisticTest Statistic

z p pp

=− 0

σz p p

p=

− 0

σ

σ pp p

n=

−0 01( )σ 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) ) >> 55


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

HH00: : pp << pp00 Reject Reject HH0 0 if if zz >> zzαα

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

Reject Reject HH0 0 if if zz << --zzα/2α/2 or or zz >> zzα/2α/2

HH00: : pp >> pp00

HH00: : pp = = pp00

Tests About a Population ProportionTests About a Population Proportion

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

Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach


Example: National Safety CouncilExample: National Safety CouncilFor a Christmas and New YearFor a Christmas and New Year’’s week, thes week, the

National Safety Council estimated thatNational Safety Council estimated that500 people would be killed and 25,000500 people would be killed and 25,000injured on the nationinjured on the nation’’s roads. Thes roads. TheNSC claimed that 50% of theNSC claimed that 50% of theaccidents would be caused byaccidents would be caused bydrunk driving.drunk driving.

TwoTwo--Tailed Test About aTailed Test About aPopulation ProportionPopulation Proportion


A sample of 120 accidents showed thatA sample of 120 accidents showed that67 were caused by drunk driving. Use67 were caused by drunk driving. Usethese data to test thethese data to test the NSCNSC’’ss claim withclaim withαα = .05.= .05.


Example: National Safety CouncilExample: National Safety Council






α α = .05= .05


0: .5H p =0: .5H p =: .5aH p ≠: .5aH p ≠

σ− −

= = =0 (67 /120) .5 1.28.045644p

p pzσ− −

= = =0 (67 /120) .5 1.28.045644p

p pz

0 0(1 ) .5(1 .5) .045644120p

p pn

σ− −

= = =0 0(1 ) .5(1 .5) .045644120p

p pn

σ− −

= = =a commona common

error is usingerror is usingin this in this

formula formula pp


pp−−Value ApproachValue Approach

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


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


For For zz = 1.28, cumulative probability = .8997= 1.28, cumulative probability = .8997pp––value = 2(1 value = 2(1 −− .8997) = .2006.8997) = .2006





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

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

Reject Reject HH00 if if zz << --1.96 or 1.96 or zz >> 1.961.96

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


End of Chapter 9End of Chapter 9

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Slides Prepared by JOHN S. LOUCKS - Cameron Universitycameron.edu/~syeda/orgl333/ch9.pdf · 2007-08-17 · © 2006 Thomson/South-Western SlideSlide 11 Slides Prepared by JOHN S. LOUCKS