1 1 Slide MA4704Gerry Golding Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether Hypothesis testing can be.

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MA4704MA4704 Gerry GoldingGerry Golding

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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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.

Example: Metro EMSExample: Metro EMS

Null and Alternative HypothesesNull and Alternative Hypotheses

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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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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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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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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..

Signal: Signal: NoiseNoise

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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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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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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

Signal: Signal: NoiseNoise

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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.

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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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..



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

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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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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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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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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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

GloGloww

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 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

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

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


= .05= .05

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

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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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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End of Chapter 9, Part AEnd of Chapter 9, Part A

1 1 Slide MA4704Gerry Golding Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether Hypothesis testing can be.

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