HYPOTHESIS TESTHYPOTHESIS TEST假设检验假设检验

InstructionInstruction

In this chapter you will learn how to In this chapter you will learn how to formulate hypotheses about a formulate hypotheses about a population mean and a population population mean and a population proportion. Through the analysis of proportion. Through the analysis of sample data, you will be able to sample data, you will be able to determine whether a hypothesis determine whether a hypothesis should or should not be rejected.should or should not be rejected.

9.1 DEVELOPING NULL AND 9.1 DEVELOPING NULL AND ALTERNATIVE HYPOTHESISALTERNATIVE HYPOTHESIS

In some application it may not be obvious how In some application it may not be obvious how the null and alternative hypothesis should be forthe null and alternative hypothesis should be formulated. We must be careful to make them propmulated. We must be careful to make them properly.erly.

There will be three types of situations in which There will be three types of situations in which hypothesis testing procedures are commonly ehypothesis testing procedures are commonly employed.mployed.

Testing Research Testing Research HypothesisHypothesis

There is an automobile model There is an automobile model example on page 325 to illustrate example on page 325 to illustrate the testing research hypothesis. the testing research hypothesis.

Testing the validity of a Testing the validity of a claim claim

there is a soft drinks example on the there is a soft drinks example on the page 325 to illustrate the testing the vpage 325 to illustrate the testing the validity of a claim. alidity of a claim.

Testing in decision-making Testing in decision-making situation situation

There is an example of shipment There is an example of shipment on the page 326 to illustrate the on the page 326 to illustrate the testing in decision-making testing in decision-making situation. situation.

SummarySummaryof Forms for Null and Alternative of Forms for Null and Alternative

HypothesesHypotheses

HH0:0: μμ≥≥µµ0 0 HH0: 0: µ µ ≤≤µµ0 0 HH0:0:µ µ = = µµ00

HHa: a: µ µ <<µµ0 0 H Ha: a: µ µ > > µµ0 0 HHa:a:µ µ ≠≠µµ00

9.2 type I and type II errors9.2 type I and type II errors

population conditionpopulation condition

----------------------------------------------------------

HH0 0 True HTrue Haa True True

Accept HAccept H0 0 Correct Type IICorrect Type II

Conclusion ErrorConclusion Error

RejectReject HH0 0 Type I CorrectType I Correct

Error ConclusionError Conclusion

InstructionsInstructions

If the sample data are consistent If the sample data are consistent with the null hypothesis Hwith the null hypothesis H0, 0, we will we will follow the practice of concluding “do follow the practice of concluding “do not reject Hnot reject H0.0.” This conclusion is ” This conclusion is preferred over “accept Hpreferred over “accept H0.0.” because ” because the conclusion to accept Hthe conclusion to accept H0 0 puts us at puts us at risk of making a Type II errorrisk of making a Type II error..

9.3 One-tailed test about a 9.3 One-tailed test about a population mean: large-sample population mean: large-sample

casecase There is a Hilltop Coffee example to illuThere is a Hilltop Coffee example to illu

strate the one-tailed test about populatistrate the one-tailed test about population mean on page 330.on mean on page 330.

The null and alternative hypotheses are The null and alternative hypotheses are as follows: as follows:

HH00:µ:µ≥≥33 HHaa:µ<3:µ<3

The test statisticThe test statistic

In the large-sample case with the population In the large-sample case with the population standard deviation standard deviation σσassumed known, the test assumed known, the test statistic is given bystatistic is given by

公式公式 : : n

xz

/

0

SummarySummary large-sample (nlarge-sample (n≥30≥30) hypothesis test about a populat) hypothesis test about a populat

ion mean for a one-tailed test of the formion mean for a one-tailed test of the form HH00:µ:µ≥≥µµ00

HHaa:µ< µ:µ< µ00

Test Statistic:Test Statistic:σσ known known Test statistic:Test statistic:σσEstimated by sEstimated by s

n

xz

/

0

ns

xz

/

0

Rejection RuleRejection Rule Using test statistic: Reject HUsing test statistic: Reject H0 0 if z<-zif z<-zαα

Using p-value: Reject HUsing p-value: Reject H0 0 if p-value<if p-value<αα

Steps of Hypothesis TestingSteps of Hypothesis TestingSteps of Hypothesis TestingSteps of Hypothesis Testing 1. Develop the null and alternative hypotheses.1. Develop the null and alternative hypotheses. 2. Specify the level of significance 2. Specify the level of significance αα.. 3. Select the statistic that will be used to test the3. Select the statistic that will be used to test the hypothesis.hypothesis.Using the test statisticUsing the test statistic 4. Use the level of significance to determine the critical v4. Use the level of significance to determine the critical v

alue for the test statistic and state the rejection rule for Halue for the test statistic and state the rejection rule for H0.0.

5. Collect the sample data and compute the value of the 5. Collect the sample data and compute the value of the test statistic.test statistic.

6. Use the value of the test statistic and the rejection rule 6. Use the value of the test statistic and the rejection rule to determine whether to reject Hto determine whether to reject H0.0.

Using the p-valueUsing the p-value 4. Collect the sample data and compute the v4. Collect the sample data and compute the v

alue of the test statistic.alue of the test statistic. 5. Use the value of the test statistic to compu5. Use the value of the test statistic to compu

te the p-value.te the p-value. 6. Reject H6. Reject H0 0 if p-value<if p-value<αα

9.4 Two-tailed test about a 9.4 Two-tailed test about a population mean: large-sample population mean: large-sample

casecase

Two-tailed tests differ from one-tailed Two-tailed tests differ from one-tailed hypothesis tests in that two-tailed tests hypothesis tests in that two-tailed tests have rejection regions in both the lower have rejection regions in both the lower and upper tails of the sampling distributand upper tails of the sampling distribution. ion.

exampleexample

There is an example of golf balls mean There is an example of golf balls mean distance at page 339.distance at page 339.

The null and alternative hypotheses arThe null and alternative hypotheses are as follows:e as follows:

HH00 : :μμ=280=280 HHaa : :μμ≠≠280280

The test statisticThe test statistic

With a sample size of 36 golf balls used each tiWith a sample size of 36 golf balls used each time the quality control test is performed, we hame the quality control test is performed, we have a large-sample case. As s result the test stative a large-sample case. As s result the test statistic is stic is

n

xz

/

0

Summary: Two-tailed tests Summary: Two-tailed tests about a population mean about a population mean

Large-sample (n≥30) hypothesis test about Large-sample (n≥30) hypothesis test about a population mean for a two-tailed test of ta population mean for a two-tailed test of the formhe form

HH00::μμ==μμ00

HHaa::μμ≠μ≠μ00

Test statistic:Test statistic:σσassumed knownassumed known

n

xz

/

0

Test statistic:Test statistic:σσ estimated by s estimated by s

Rejection ruleRejection rule Using test statistic: Reject HUsing test statistic: Reject H00 if z<-z if z<-zαα

/2/2 or if z>z or if z>zαα/2/2

Using p-value: Reject HUsing p-value: Reject H0 0 if p-if p-value<value<αα

ns

xz

/

0

Relationship between interval Relationship between interval estimation and hypothesis estimation and hypothesis

testing testing

The details about the differences betweThe details about the differences between these two approaches are at page 34en these two approaches are at page 342-343. 2-343.

A confidence interval approach A confidence interval approach to testing a hypothesis of the to testing a hypothesis of the

formform HH00::μμ==μμ00

HHaa::μμ≠μ≠μ00

1.Select a simple random sample from the popul1.Select a simple random sample from the population and use the value of the sample mean ation and use the value of the sample mean to develop the confidence interval for the popto develop the confidence interval for the population mean ulation mean μμ. If . If σσis assumed known, compuis assumed known, compute the interval estimate by using te the interval estimate by using

x

nzx

2

If If σisσis estimated by s, compute the interval estimated by s, compute the interval estimate by using estimate by using

2. If the confidence interval contains the h2. If the confidence interval contains the hypothesized value ypothesized value μμ00, do not reject H, do not reject H00..

otherwise, reject Hotherwise, reject H00..

n

szx 2

9.5 Test about a population 9.5 Test about a population mean: small-sample casemean: small-sample case

The procedures for hypothesis test aboThe procedures for hypothesis test about a population mean discussed in sectiut a population mean discussed in section 9.3 and 9.4 were based on the central on 9.3 and 9.4 were based on the central limit theorem and large-sample theory.limit theorem and large-sample theory.

In section 9.5, we consider tests about In section 9.5, we consider tests about a population mean using a small sample a population mean using a small sample (n<30).(n<30).

AssumptionAssumption

The small-sample case require the assuThe small-sample case require the assumption that the population has a mption that the population has a normanormall probability distribution. probability distribution.

If this assumption is not appropriate, tIf this assumption is not appropriate, the best alternative is to increase the sahe best alternative is to increase the sample size to n≥30 and rely on the large-mple size to n≥30 and rely on the large-sample hypothesis testing procedures.sample hypothesis testing procedures.

Test statisticTest statistic

If the population standard deviation If the population standard deviation σσ is ass is assumed known:umed known:

If the population standard deviation If the population standard deviation σσis estiis estimated by the sample standard deviation s:mated by the sample standard deviation s:

n

xz

0

nsx

t0

There is an example of the internationaThere is an example of the international air transport association on page 348.l air transport association on page 348.

p-values and the t distribution is on pap-values and the t distribution is on page 348-349.ge 348-349.

Two-tailed test is similar to one-tailed tTwo-tailed test is similar to one-tailed test. It is on the page 349-350.est. It is on the page 349-350.

9.6 Test about a population 9.6 Test about a population proportionproportion

Three forms for a hypothesis test Three forms for a hypothesis test about a population proportion are as about a population proportion are as follows:follows:

HH00:p:p≥≥pp0 0 HH00:p:p≤p≤p0 0 HH00:p=p:p=p00

HHa a :p<p:p<p0 0 HHa a :p>p:p>p0 0 HHaa:p:p≠p≠p00

The first two forms are one-tailed The first two forms are one-tailed tests, and the third form is two-tailed tests, and the third form is two-tailed test. test.

If both np and n(1-p) are greater than or equaIf both np and n(1-p) are greater than or equal to 5. The sampling distribution of can be al to 5. The sampling distribution of can be approximated by a normal probability distributipproximated by a normal probability distribution , the following test statistic can be used.on , the following test statistic can be used.

Test statistic for test about a population propTest statistic for test about a population proportion:ortion:

wherewhere

p

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0

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ppp

)1( 00

9.7 hypothesis testing and 9.7 hypothesis testing and decision makingdecision making

There is an example of shipment of batThere is an example of shipment of batteries on page 360.teries on page 360.

When we do not reject HWhen we do not reject H0, 0, action we taaction we take may risk Type ke may risk Type ІІІІ error . In section 9.8 a error . In section 9.8 and 9.9 we explain how to compute the prnd 9.9 we explain how to compute the probability of making a Type obability of making a Type ІІІІ error and h error and how the sample size can be adjusted to heow the sample size can be adjusted to help control the probability of making a Tylp control the probability of making a Type pe ІІІІ error. error.

9.8 calculating the probability of t9.8 calculating the probability of type ype ІІІІ errors errors

There is an example on page 360-362. There is an example on page 360-362. suppose suppose αα=.05 and a mean life of µ=112=.05 and a mean life of µ=112

hours.hours. probability of a Type probability of a Type ІІІІ error when µ=112 error when µ=112

ββ=.0091=.0091 112 116.71 112 116.71 Accept HAccept H00

There is a table to show the probability of making a Type There is a table to show the probability of making a Type ІІІІ error error for a variety of values of µ less than 120. Note that as µ increases for a variety of values of µ less than 120. Note that as µ increases toward 120, the probability of making a Type toward 120, the probability of making a Type ІІІІ error increases to error increases toward an upper bound of .95.ward an upper bound of .95.

value of µ value of µ 公式 公式 p ability of power p ability of power a Type a Type ІІІІ error( error(ββ) (1-) (1-ββ)) 112 2.36 .0091 .9909112 2.36 .0091 .9909 114 1.36 .0869 .9131114 1.36 .0869 .9131 115 .86 .1949 .8051115 .86 .1949 .8051 116.17 .00 .5000 .5000116.17 .00 .5000 .5000 117 -.15 .5596 .4404117 -.15 .5596 .4404 118 -.65 .7422 .2578118 -.65 .7422 .2578 119.999 -.1.645 .9500 .0500119.999 -.1.645 .9500 .0500

Power curve for the lot-Power curve for the lot-acceptance hypothesis testacceptance hypothesis test

1.001.00

.80.80

.60.60

.40.40

.20.20

112 115 118 120112 115 118 120

summarysummary Procedure to compute the probability of making a Type Procedure to compute the probability of making a Type ІІІІ error: error: 1. Formulate the null and alternative hypothesis1. Formulate the null and alternative hypothesis 2. Use the level of significance 2. Use the level of significance ααto establish a rejection rule based to establish a rejection rule based

on the test statistic.on the test statistic. 3. Using the rejection rule, solve for the value of the sample mean 3. Using the rejection rule, solve for the value of the sample mean

that identifies the rejection region for the test.that identifies the rejection region for the test. 4. Use the results from step3 to state the values of the sample me4. Use the results from step3 to state the values of the sample me

an that lead to the acceptance of Han that lead to the acceptance of H00. It also defines the acceptan. It also defines the acceptance region for the test.ce region for the test.

5. Using the sampling distribution of for any value of µ from the 5. Using the sampling distribution of for any value of µ from the alternative hypothesis, and the acceptance region from step4, calternative hypothesis, and the acceptance region from step4, compute the probability that the sample mean will be in the acceompute the probability that the sample mean will be in the acceptance region. This probability is the probability of making a Typtance region. This probability is the probability of making a Type pe ІІІІ error at the chosen value of µ error at the chosen value of µ

x

9.9 determining the sample size for 9.9 determining the sample size for a hypothesis test about a population a hypothesis test about a population

meanmean Recommended sample size for a one-tailed hypothesis test about Recommended sample size for a one-tailed hypothesis test about

a population meana population mean

WhereWhere zzαα=z value providing an area of =z value providing an area of ααin the tail of a standard normal in the tail of a standard normal

distributiondistribution zzββ=z value providing an area of =z value providing an area of ββin the tail of a standard normal in the tail of a standard normal

distributiondistribution σσ=the population standard deviation=the population standard deviation µµ0 0 =the value of the population mean in the null hypothesis=the value of the population mean in the null hypothesis µa µa =the value of the population mean used for the Type =the value of the population mean used for the Type ІІІІ error error

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