1 1 slide © 2006 thomson/south-western chapter 9 hypothesis testing developing null and alternative...

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© 2006 Thomson/South-Western© 2006 Thomson/South-Western

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 isThe alternative hypothesis is usually usually what the test is what the test is attempting to establish.attempting to establish.

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Testing in Decision-Making SituationsTesting in Decision-Making Situations

Developing Null and Alternative Developing Null and Alternative HypothesesHypotheses

• 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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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 what is expected under the null. Select Specify what is expected under the null. Select aa . .

Step 3.Step 3. Collect the sample data and compute Collect the sample data and compute the test statistic to compare observed the test statistic to compare observed and expected results.and expected results.

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 OR choose the critical value-value OR choose the critical value

Step 5.Step 5. Reject Reject HH00 if if pp-value -value << (chosen sig. (chosen sig. level) level) OR Reject OR Reject HH0 if test stat. > critical 0 if test stat. > critical valuevalue

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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

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.

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Example: Metro EMSExample: Metro EMS

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.

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

KnownKnown

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Null and Alternative HypothesesNull and Alternative Hypotheses

The emergency service is meetingThe emergency service is meeting

the response goal; no follow-upthe response goal; no follow-up

action is necessary.action is necessary.

The emergency service is notThe emergency service is not

meeting the response goal;meeting the response goal;

appropriate follow-up action isappropriate follow-up action is

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

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

12 minutes.12 minutes.

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

Critical Value ApproachCritical Value Approach


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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00 z = 1.645 z = 1.645

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

zz

Samplingdistribution

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

12 minutes.12 minutes.

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..

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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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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

GloGloww

HH00: :

HHaa:: 6 6

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

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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Example: Glow ToothpasteExample: Glow Toothpaste

Two-Tailed Test About a Population Mean: Two-Tailed Test About a Population Mean: Known Known

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.

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Example: Glow ToothpasteExample: Glow Toothpaste

Two-Tailed Test About a Population Mean: Two-Tailed Test About a Population Mean: KnownKnown

oz.

GlowGlow Perform 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.

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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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GloGloww


KnownKnown


We are at least 97% confident that the We are at least 97% confident that the mean filling weight of the toothpaste mean filling weight of the toothpaste

tubes is not 6 oz.tubes is not 6 oz.


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

GloGloww



of


of z xn

0

/z x

n

0

/


KnownKnown

/2 = .015/2 = .015

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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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GloGloww


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..

We are at least 97% confident that the We are at least 97% confident that the mean filling weight of the toothpaste mean filling weight of the toothpaste

tubes is not 6 oz.tubes is not 6 oz.

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GloGloww


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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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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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 samplesA State Highway Patrol periodically samples

vehicle speeds at various locationsvehicle speeds at various locations

on a particular roadway. on a particular roadway.

The sample of vehicle speedsThe sample of vehicle speeds

is used to test the hypothesisis used to test the hypothesis

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 deemed is rejected are deemed

the best locations for radar traps.the best locations for radar traps.

HH00: : << 65 65

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Example: Highway PatrolExample: Highway Patrol

One-Tailed Test About a Population Mean: One-Tailed Test About a Population Mean: UnknownUnknown At 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 a

standard deviation ofstandard deviation of

4.2 mph. Use 4.2 mph. Use = .05 to = .05 to

test the hypothesis.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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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

Do Not Reject H0Do Not Reject H0

tt


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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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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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= .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

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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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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