1 1 Slide © 2005 Thomson/South-Western Chapter 9, Part B Hypothesis Tests Population Proportion...

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

Chapter 9, Part BChapter 9, Part B Hypothesis Tests Hypothesis Tests

Population ProportionPopulation Proportion Hypothesis Testing and Decision MakingHypothesis Testing and Decision Making Calculating the Probability of Type II ErrorsCalculating the Probability of Type II Errors Determining the Sample Size forDetermining the Sample Size for Hypothesis Tests About a Population MeanHypothesis Tests About a Population Mean

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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 CouncilExample: National Safety Council

For a Christmas and New Year’s week, theFor a Christmas and New Year’s week, the

National Safety Council estimated thatNational Safety Council estimated that

500 people would be killed and 25,000500 people would be killed and 25,000

injured on the nation’s roads. Theinjured on the nation’s roads. The

NSC claimed that 50% of theNSC claimed that 50% of the

accidents would be caused byaccidents would be caused by

drunk driving.drunk driving.

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

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A sample of 120 accidents showed thatA sample of 120 accidents showed that

67 were caused by drunk driving. Use67 were caused by drunk driving. Use

these data to test the NSC’s claim withthese data to test the NSC’s claim with

= .05.= .05.


Example: National Safety CouncilExample: National Safety Council

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

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

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

= .05= .05

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

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

pp

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ppValue ApproachValue Approach

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

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

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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Critical Value ApproachCritical Value Approach

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

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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Hypothesis Testing and Decision MakingHypothesis Testing and Decision Making

In many decision-making situations the decisionIn many decision-making situations the decision maker may want, and in some cases may be forced,maker may want, and in some cases may be forced, to take action with both the conclusion do not reject to take action with both the conclusion do not reject

HH0 0 and the conclusion reject and the conclusion reject HH00..

In such situations, it is recommended that theIn such situations, it is recommended that the hypothesis-testing procedure be extended to hypothesis-testing procedure be extended to

includeinclude consideration of making a Type II error.consideration of making a Type II error.

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Calculating the Probability of a Type II Calculating the Probability of a Type II Error Error

in Hypothesis Tests About a Population in Hypothesis Tests About a Population MeanMean1.1. Formulate the null and alternative hypotheses. Formulate the null and alternative hypotheses.

3.3. Using the rejection rule, solve for the value of the Using the rejection rule, solve for the value of the sample mean corresponding to the critical value ofsample mean corresponding to the critical value of the test statistic.the test statistic.

2.2. Using the critical value approach, use the level of Using the critical value approach, use the level of significance significance to determine the critical value and to determine the critical value and the rejection rule for the test.the rejection rule for the test.

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Calculating the Probability of a Type II Calculating the Probability of a Type II Error Error

in Hypothesis Tests About a Population in Hypothesis Tests About a Population MeanMean4.4. Use the results from step 3 to state the values of the Use the results from step 3 to state the values of the

sample mean that lead to the acceptance of sample mean that lead to the acceptance of HH00; this ; this defines the acceptance region.defines the acceptance region.

5.5. Using the sampling distribution of for a value of Using the sampling distribution of for a value of satisfying the alternative hypothesis, and the acceptancesatisfying the alternative hypothesis, and the acceptance region from step 4, compute the probability that theregion from step 4, compute the probability that the sample mean will be in the acceptance region. (This issample mean will be in the acceptance region. (This is the probability of making a Type II error at the chosenthe probability of making a Type II error at the chosen level of level of .).)

xx

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Example: Metro EMS (revisited)Example: Metro EMS (revisited)

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 or not the service goal of 12 minutes or less iswhether or not the service goal of 12 minutes or less isbeing achieved.being achieved.

Recall that the response times forRecall that the response times fora random sample of 40 medicala random sample of 40 medicalemergencies were tabulated. Theemergencies were tabulated. Thesample mean is 13.25 minutes.sample mean is 13.25 minutes.The population standard deviationThe population standard deviationis believed to be 3.2 minutes.is believed to be 3.2 minutes.

Calculating the ProbabilityCalculating the Probability of a Type II Error of a Type II Error

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121.645

3.2/ 40x

z

12

1.6453.2/ 40

xz

3.212 1.645 12.8323

40x

3.212 1.645 12.8323

40x

4.4. We will accept We will accept HH00 when when xx < 12.8323 < 12.8323

3.3. Value of the sample mean that identifies Value of the sample mean that identifies the rejection region:the rejection region:

2.2. Rejection rule is: Reject Rejection rule is: Reject HH00 if if zz >> 1.645 1.6451.1. Hypotheses are: Hypotheses are: HH00: : and and HHaa::


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12.0001 1.645 .9500 .050012.0001 1.645 .9500 .050012.4 0.85 .8023 .197712.4 0.85 .8023 .197712.8 0.06 .5239 .476112.8 0.06 .5239 .476112.8323 0.00 .5000 .500012.8323 0.00 .5000 .500013.2 -0.73 .2327 .767313.2 -0.73 .2327 .767313.6 -1.52 .0643 .935713.6 -1.52 .0643 .935714.0 -2.31 .0104 .989614.0 -2.31 .0104 .9896

12.83233.2/ 40

z

12.83233.2/ 40

z

Values of Values of 1-1-

5.5. Probabilities that the sample mean will be Probabilities that the sample mean will be in the acceptance region:in the acceptance region:


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Calculating the Probability of a Type II ErrorCalculating the Probability of a Type II Error

When the true population mean When the true population mean is far is far aboveabove the null hypothesis value of 12, there is a the null hypothesis value of 12, there is a lowlow probability that we will make a Type II probability that we will make a Type II error.error.

When the true population mean When the true population mean is close to is close to the null hypothesis value of 12, there is a highthe null hypothesis value of 12, there is a high probability that we will make a Type II error.probability that we will make a Type II error.

Observations about the preceding table:Observations about the preceding table:

Example: Example: = 12.0001, = 12.0001, = .9500 = .9500

Example: Example: = 14.0, = 14.0, = .0104 = .0104

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Power of the TestPower of the Test

The probability of correctly rejecting The probability of correctly rejecting HH00 when it is when it is false is called the false is called the powerpower of the test. of the test.

For any particular value of For any particular value of , the power is 1 – , the power is 1 – ..

We can show graphically the power associatedWe can show graphically the power associated with each value of with each value of ; such a graph is called a; such a graph is called a power curvepower curve. (See next slide.). (See next slide.)

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Power CurvePower Curve

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

11.5 12.0 12.5 13.0 13.5 14.0 14.5

Pro

bab

ilit

y of

Cor

rect

lyR

ejec

ting

Null

Hyp

othes

is

H0 False

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The specified level of significance determines theThe specified level of significance determines the probability of making a Type I error.probability of making a Type I error.

By controlling the sample size, the probability ofBy controlling the sample size, the probability of making a Type II error is controlled.making a Type II error is controlled.

Determining the Sample Size for a Determining the Sample Size for a Hypothesis Test About a Population MeanHypothesis Test About a Population Mean

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00

aa

xx

xx

Samplingdistribution of whenH0 is trueand = 0

Samplingdistribution of whenH0 is trueand = 0

xx

Samplingdistribution of whenH0 is falseand a > 0

Samplingdistribution of whenH0 is falseand a > 0

xx

Reject H0Reject H0

cc

cc

HH00: :

HHaa::

x n

x nNote:Note:


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wherewhere

zz = = zz value providing an area of value providing an area of in the tail in the tail

zz = = zz value providing an area of value providing an area of in the tail in the tail= population standard deviation= population standard deviation

00 = value of the population mean in = value of the population mean in HH00

a a = value of the population mean used for = value of the population mean used for thethe Type II error Type II error

nz z

a

( )

( )

2 2

02

nz z

a

( )

( )

2 2

02


Note: In a two-tailed hypothesis test, use Note: In a two-tailed hypothesis test, use zz /2 /2 not not zz

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Relationship Among Relationship Among , , , and , and nn

Once two of the three values are known, the Once two of the three values are known, the other can be computed.other can be computed.

For a given level of significance For a given level of significance , increasing , increasing the sample size the sample size nn will reduce will reduce ..

For a given sample size For a given sample size nn, decreasing , decreasing will will increase increase , whereas increasing , whereas increasing will decrease will decrease b.b.

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Let’s assume that the director of medicalLet’s assume that the director of medical services makes the following statements about theservices makes the following statements about the

allowable probabilities for the Type I and Type IIallowable probabilities for the Type I and Type II errors:errors:

•If the mean response time is If the mean response time is = 12 minutes, I = 12 minutes, I am willing to risk an am willing to risk an = .05 probability of = .05 probability of rejecting rejecting HH00..•If the mean response time is 0.75 minutes over If the mean response time is 0.75 minutes over the specification (the specification ( = 12.75), I am willing to risk = 12.75), I am willing to risk a a = .10 probability of not rejecting = .10 probability of not rejecting HH00..

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= .05, = .05, = .10 = .10

zz = 1.645, = 1.645, zz = 1.28= 1.28

00 = 12, = 12, a a = 12.75= 12.75= 3.2= 3.2

2 2 2 2

2 20

( ) (1.645 1.28) (3.2)155.75 156

( ) (12 12.75)a

z zn

2 2 2 2

2 20

( ) (1.645 1.28) (3.2)155.75 156

( ) (12 12.75)a

z zn

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End of Chapter 9, Part BEnd of Chapter 9, Part B

1 1 Slide © 2005 Thomson/South-Western Chapter 9, Part B Hypothesis Tests Population Proportion...

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