1 1 slide statistics for business and economics seventh edition andersonsweeneywilliams slides...

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STATISTICS FORSTATISTICS FORBUSINESS AND ECONOMICSBUSINESS AND ECONOMICSSeventh EditionSeventh Edition

AndersonAnderson

Sweeney Sweeney

WilliamsWilliams

Slides Prepared bySlides Prepared by John LoucksJohn Loucks

© 1999 ITP/South-Western College Publishing© 1999 ITP/South-Western College Publishing

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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 Tests About a Population Mean:Tests About a Population Mean:

Large-Sample CaseLarge-Sample Case Tests About a Population Mean:Tests About a Population Mean:

Small-Sample CaseSmall-Sample Case Tests About a Population ProportionTests About a Population 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 for a HypothesisDetermining the Sample Size for a Hypothesis

Test About a Population MeanTest About a Population Mean

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Developing Null and Alternative Developing Null and Alternative HypothesesHypotheses

Hypothesis testingHypothesis testing can be used to determine can be used to determine whether a statement about the value of a population whether a statement about the value of a population parameter should or should not be rejected.parameter should or should not be rejected.

The The null hypothesisnull hypothesis, , denoted by denoted by HH0 0 , , is a tentative is a tentative assumption about a population parameter.assumption about a population parameter.

The The alternative hypothesisalternative hypothesis, denoted by , denoted by HHa a , is the , is the

opposite of what is stated in the null hypothesis.opposite of what is stated in the null hypothesis. Hypothesis testing is similar to a criminal trial. The Hypothesis testing is similar to a criminal trial. The

hypotheses are:hypotheses are:

HH00: The defendant is innocent: The defendant is innocent

HHaa: The defendant is guilty: The defendant is guilty

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Testing Research HypothesesTesting Research Hypotheses• The research hypothesis should be expressed The research hypothesis should be expressed

as the alternative hypothesis.as the alternative hypothesis.• The conclusion that the research hypothesis is The conclusion that the research hypothesis is

true comes from sample data that contradict true comes from sample data that contradict the null hypothesis.the null hypothesis.

Testing the Validity of a ClaimTesting the Validity of a Claim• Manufacturers’ claims are usually given the Manufacturers’ claims are usually given the

benefit of the doubt and stated as the null benefit of the doubt and stated as the null hypothesis.hypothesis.

• The conclusion that the claim is false comes The conclusion that the claim is false comes from sample data that contradict the null from sample data that contradict the null hypothesis.hypothesis.


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Testing in Decision-Making SituationsTesting in Decision-Making Situations• A decision maker might have to choose A decision maker might have to choose

between two courses of action, one between two courses of action, one associated with the null hypothesis and associated with the null hypothesis and another associated with the alternative another associated with the alternative hypothesis.hypothesis.

• Example: Accepting a shipment of goods Example: Accepting a shipment of goods from a supplier or returning the shipment of from a supplier or returning the shipment of goods to the supplier.goods to the supplier.


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A Summary of Forms for Null and A Summary of Forms for Null and Alternative Hypotheses about a Alternative Hypotheses about a

Population MeanPopulation Mean The equality part of the hypotheses always appears The equality part of the hypotheses always appears

in the null hypothesis.in the null hypothesis. In general, a hypothesis test about the value of a In general, a hypothesis test about the value of a

population mean population mean must take one of the following must 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).

HH00: : >> 00 HH00: : << 0 0 HH00: : = = 00

HHaa: : < < 00 HHaa: : > > 00 HHaa: : 00

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Example: Metro EMSExample: Metro EMS

A major west coast city provides one of the mostA major west coast city provides one of the most

comprehensive emergency medical services in thecomprehensive emergency medical services in the

world. Operating in a multiple hospital system withworld. Operating in a multiple hospital system with

approximately 20 mobile medical units, the service goalapproximately 20 mobile medical units, the service goal

is to respond to medical emergencies with a mean timeis to respond to medical emergencies with a mean time

of 12 minutes or less.of 12 minutes or less.

The director of medical services wants to formulate aThe director of medical services wants to formulate a

hypothesis test that could use a sample of emergencyhypothesis test that could use a sample of emergency

response times to determine whether or not the serviceresponse times to determine whether or not the service

goal of 12 minutes or less is being achieved.goal of 12 minutes or less is being achieved.

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HypothesesHypotheses Conclusion and ActionConclusion and Action

HH00: : The emergency service is meeting The emergency service is meeting

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

action is necessary.action is necessary.

HHaa:: The emergency service is not The emergency service is not

meeting the response goal;meeting the response goal;

appropriate follow-up action isappropriate follow-up action is

necessary.necessary.

wherewhere

= mean response time for the population= mean response time for the population

of medical emergency requests.of medical emergency requests.

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Type I and Type II ErrorsType I and Type II Errors Since hypothesis tests are based on sample data, we Since hypothesis tests are based on sample data, we

must allow for the possibility of errors.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.

A A Type II errorType II error is accepting is accepting HH00 when it is false. when it is false. The person conducting the hypothesis test specifies The person conducting the hypothesis test specifies

the maximum allowable probability of making athe maximum allowable probability of making a

Type I error, denoted by Type I error, denoted by and called the and called the level of level of significancesignificance..

Generally, we cannot control for the probability of Generally, we cannot control for the probability of making a Type II error, denoted by making a Type II error, denoted by ..

Statisticians avoid the risk of making a Type II error by Statisticians avoid the risk of making a Type II error by using “do not reject 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

Population ConditionPopulation Condition

HH0 0 TrueTrue HHa a TrueTrue

ConclusionConclusion (( ) ) (( ) )

AcceptAccept HH00 CorrectCorrect Type II Type II

(Conclude (Conclude ConclusionConclusion Error Error

RejectReject HH00 Type IType I Correct Correct

(Conclude (Conclude rrorrror Conclusion Conclusion

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One-Tailed Tests About a Population One-Tailed Tests About a Population Mean: Large-Sample Case (Mean: Large-Sample Case (nn >> 30) 30)

Hypotheses:Hypotheses: HH00: : oror HH00: :

HHaa::HHaa::

Test Statistic:Test Statistic: KnownKnown Unknown Unknown

Rejection Rule:Rejection Rule:

Reject Reject HH0 0 if if zz > > zzReject Reject HH0 0

if if zz < - < -zz

z xn

0

/z x

n

0

/z x

s n 0/

z xs n

0/

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One-Tailed Test about a Population MeanOne-Tailed Test about a Population Mean

Let Let = = P P (Type I Error) = .05 (Type I Error) = .05

Sampling distribution of (assuming H0 is true and = 12)


xx

1212 c c

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

xx1.6451.645xx

(Critical value)(Critical value)

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One-Tailed Test about a Population MeanOne-Tailed Test about a Population Mean

Let Let nn = 40, = 13.25 minutes, = 40, = 13.25 minutes, ss = 3.2 minutes = 3.2 minutes

(The sample standard deviation (The sample standard deviation ss can be used to can be used to

estimate the population standard deviation estimate the population standard deviation .).)

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

ConclusionConclusion: : We are 95% confident that Metro EMSWe are 95% confident that Metro EMS

is not meeting the response goal of 12 minutes;is not meeting the response goal of 12 minutes;

appropriate action should be taken to improveappropriate action should be taken to improve

service.service.

xx

zx

n

/.. /

.13 25 123 2 40

2 47zx

n

/.. /

.13 25 123 2 40

2 47

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The Use of The Use of p p -Values-Values

The The p p -value is the probability of obtaining a -value is the probability of obtaining a sample result that is at least as unlikely as what is sample result that is at least as unlikely as what is observed.observed.

The The p p -value can be used to make the decision in a -value can be used to make the decision in a hypothesis test by noting that:hypothesis test by noting that:• if the if the p p -value is less than the level of -value is less than the level of

significance significance , the value of the test statistic is in , the value of the test statistic is in the rejection region.the rejection region.

• if the if the p p -value is greater than or equal to -value is greater than or equal to , the , the value of the test statistic is not in the rejection value of the test statistic is not in the rejection region.region.

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

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Using the Using the p p -value to Test the Hypothesis-value to Test the Hypothesis

Recall that Recall that zz = 2.47 for = 13.25. Then = 2.47 for = 13.25. Then p p --value = .0068. Since value = .0068. Since p p -value < -value < , that is .0068 , that is .0068 < .05, we reject< .05, we reject HH00.

Using the Using the p p -value to Test the Hypothesis-value to Test the Hypothesis

Recall that Recall that zz = 2.47 for = 13.25. Then = 2.47 for = 13.25. Then p p --value = .0068. Since value = .0068. Since p p -value < -value < , that is .0068 , that is .0068 < .05, we reject< .05, we reject HH00.

xx

p p -value-valuep p -value-value

00 1.645 1.645

Do Not Reject H0Do Not Reject H0

Reject H0Reject H0

zz2.472.47

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The Steps of Hypothesis TestingThe Steps of Hypothesis Testing

Determine the appropriate hypotheses.Determine the appropriate hypotheses. Select the test statistic for deciding whether or Select the test statistic for deciding whether or

not to reject the null hypothesis.not to reject the null hypothesis. Specify the level of significance Specify the level of significance for the test. for the test. Use Use to develop the rule for rejecting to develop the rule for rejecting HH00.. Collect the sample data and compute the value Collect the sample data and compute the value

of the test statistic.of the test statistic. (a) Compare the test statistic to the critical (a) Compare the test statistic to the critical

value(s) in the rejection rule, or (b) Compute the value(s) in the rejection rule, or (b) Compute the p p -value based on the test statistic and compare -value based on the test statistic and compare it to it to to determine whether or not to reject to determine whether or not to reject HH00..

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Two-Tailed Tests About a Population Two-Tailed Tests About a Population Mean: Large-Sample Case (Mean: Large-Sample Case (nn >> 30) 30)

Hypotheses:Hypotheses: HH00: :

HHaa::



Reject Reject HH0 0 if |if |zz| > | > zz

z xn

0

/z x

n

0

/z x

s n 0/

z xs n

0/

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Example: Glow ToothpasteExample: Glow Toothpaste

The production line for Glow toothpaste is designedThe production line for Glow toothpaste is designed

to fill tubes of toothpaste with a mean weight of 6to fill tubes of toothpaste with a mean weight of 6

ounces. Data available show that the weight has aounces. Data available show that the weight has a

standard deviation of .2 ounces.standard deviation of .2 ounces.

Periodically, a sample of 30 tubes will be selected inPeriodically, a sample of 30 tubes will be selected in

order to check the filling process. Quality assuranceorder to check the filling process. Quality assurance

procedures call for the continuation of the fillingprocedures call for the continuation of the filling

process if the sample results are consistent with theprocess if the sample results are consistent with the

assumption that the mean filling weight for theassumption that the mean filling weight for the

population of toothpaste tubes is 6 ounces; otherwise population of toothpaste tubes is 6 ounces; otherwise

the filling process will be stopped and adjusted.the filling process will be stopped and adjusted.

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A hypothesis test about the population mean can beA hypothesis test about the population mean can be

used to help determine when the filling process shouldused to help determine when the filling process should

continue operating and when it should be stopped andcontinue operating and when it should be stopped and

corrected.corrected.

The hypotheses are:The hypotheses are:

HH00: :

HHaa::

With a .05 level of significance, the decision rule is:With a .05 level of significance, the decision rule is:

Reject Reject HH0 0 if if z z < -1.96 or if < -1.96 or if zz > 1.96 > 1.96

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Two-Tailed Test about a Population MeanTwo-Tailed Test about a Population Mean



xx

0 0 1.96 1.96

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

-1.96 -1.96

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Assume that a sample of 30 toothpaste tubesAssume that a sample of 30 toothpaste tubes

provides a sample mean of 6.1 ounces.provides a sample mean of 6.1 ounces.

Let Let nn = 30, = 6.1 ounces, = 30, = 6.1 ounces, = .2 ounces = .2 ounces


Conclusion: Conclusion: We are 95% confident that the mean We are 95% confident that the mean

filling weight of the toothpaste tubes is not 6 filling weight of the toothpaste tubes is not 6 ounces. The filling process should be stopped and ounces. The filling process should be stopped and the filling mechanism adjusted.the filling mechanism adjusted.


Assume that a sample of 30 toothpaste tubesAssume that a sample of 30 toothpaste tubes

provides a sample mean of 6.1 ounces.provides a sample mean of 6.1 ounces.

Let Let nn = 30, = 6.1 ounces, = 30, = 6.1 ounces, = .2 ounces = .2 ounces


Conclusion: Conclusion: We are 95% confident that the mean We are 95% confident that the mean

filling weight of the toothpaste tubes is not 6 filling weight of the toothpaste tubes is not 6 ounces. The filling process should be stopped and ounces. The filling process should be stopped and the filling mechanism adjusted.the filling mechanism adjusted.

zx

n

0 6 1 6

2 302 74

/.

. /.z

xn

0 6 1 62 30

2 74/

.. /

.

xx

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Using the Using the p p -Value for a Two-Tailed Hypothesis Test-Value for a Two-Tailed Hypothesis Test

Suppose we define the Suppose we define the p p -value for a two-tailed test as -value for a two-tailed test as double double the area found in the tail of the distribution.the area found in the tail of the distribution.

With With zz = 2.74, the standard normal probability = 2.74, the standard normal probability

table shows there is a .5000 - .4969 = .0031 probabilitytable shows there is a .5000 - .4969 = .0031 probability

of a difference larger than .1 in the upper tail of theof a difference larger than .1 in the upper tail of the

distribution.distribution.

Considering the same probability of a larger difference in Considering the same probability of a larger difference in the lower tail of the distribution, we havethe lower tail of the distribution, we have

p p -value = 2(.0031) = .0062-value = 2(.0031) = .0062

The The p p -value .0062 is less than -value .0062 is less than = .05, so = .05, so HH00 is rejected. is rejected.

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Confidence Interval Approach to aConfidence Interval Approach to aTwo-Tailed Test about a Population MeanTwo-Tailed Test about a Population Mean

Select a simple random sample from the Select a simple random sample from the population and use the value of the sample population and use the value of the sample mean to develop the confidence interval for mean to develop the confidence interval for the population mean the population mean ..

If the confidence interval contains the If the confidence interval contains the hypothesized value hypothesized value 00, do not reject , do not reject HH00. . Otherwise, reject Otherwise, reject HH00 . .

xx

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Confidence Interval Approach to a Two-Tailed Confidence Interval Approach to a Two-Tailed Hypothesis TestHypothesis Test

The 95% confidence interval for The 95% confidence interval for is is

or or 6.0284 to 6.17166.0284 to 6.1716

Since the hypothesized value for the population Since the hypothesized value for the population mean, mean, 00 = 6, is not in this interval, the hypothesis- = 6, is not in this interval, the hypothesis-testing conclusion is that the null hypothesis,testing conclusion is that the null hypothesis,

HH00: : = 6, can be rejected. = 6, can be rejected.

x zn

/ . . (. ) . .2 6 1 1 96 2 30 6 1 0716x zn

/ . . (. ) . .2 6 1 1 96 2 30 6 1 0716

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Tests About a Population Mean:Tests About a Population Mean:Small-Sample Case (Small-Sample Case (nn < 30) < 30)


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

Rejection Rule: Rejection Rule:

One-TailedOne-Tailed Two-TailedTwo-Tailed

HH00: : Reject Reject HH0 0 if if tt > > tt

HH00: : Reject Reject HH0 0 if if tt < - < -tt

HH00: : Reject Reject HH0 0 if |if |tt| > | > tt

tx

n

0

/tx

n

0

/txs n

0/

txs n

0/

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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 distribution table provided in most statistics textbooks does not have in most statistics textbooks does not have sufficient detail to determine the sufficient detail 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 distribution table to identify a table to identify a rangerange for the for the p p -value.-value.

An advantage of computer software packages An advantage of computer software packages is that the computer output will provide the is that the computer output will provide the p p --value for thevalue for the

tt distribution. distribution.

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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 The equality part of the hypotheses always The equality part of the hypotheses always

appears in the null hypothesis.appears in the null hypothesis. In general, a hypothesis test about the value of a In general, a hypothesis test about the value of a

population proportion population proportion pp must take one of the must take one of the following three forms (where following three forms (where pp00 is the is the hypothesized value of the population proportion). hypothesized value of the population proportion).

HH00: : pp >> pp00 HH00: : pp << pp00 HH00: : pp = = pp00

HHaa: : pp < < pp00 HHaa: : pp > > pp0 0 HHaa: : pp pp00

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Tests About a Population Proportion:Tests About a Population Proportion:Large-Sample Case (Large-Sample Case (npnp >> 5 and 5 and n n (1 - (1 - pp) )

>> 5) 5) Test Statistic:Test Statistic:

wherewhere


One-TailedOne-Tailed Two-TailedTwo-Tailed

HH00: : pppp Reject Reject HH0 0 if z > zif z > z

HH00: : pppp Reject Reject HH0 0 if z < -zif z < -z

HH00: : pppp Reject Reject HH0 0 if |z| > if |z| > zz

zp p

p

0

z

p p

p

0

pp p

n

0 01( ) pp p

n

0 01( )

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Hypothesis Testing and Decision MakingHypothesis Testing and Decision Making

In many decision-making situations the In many decision-making situations the decision maker may want, and in some cases decision maker may want, and in some cases may be forced, to take action with both the may be forced, to take action with both the conclusion do not reject conclusion do not reject HH00 and the conclusion and the conclusion reject reject HH00..

In such situations, it is recommended that the In such situations, it is recommended that the hypothesis-testing procedure be extended to hypothesis-testing procedure be extended to include consideration of making a Type II error.include 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. Formulate the null and alternative hypotheses.1. Formulate the null and alternative hypotheses.

2. Use the level of significance 2. Use the level of significance to establish a to establish a rejection rule based on the test statistic.rejection rule based on the test statistic.

3. Using the rejection rule, solve for the value of the 3. Using the rejection rule, solve for the value of the sample mean that identifies the rejection region.sample mean that identifies the rejection region.

4. Use the results from step 3 to state the values of 4. Use the results from step 3 to state the values of the sample mean that lead to the acceptance of the sample mean that lead to the acceptance of HH00; ; this defines the acceptance region.this defines the acceptance region.

5. Using the sampling distribution of for any value of 5. Using the sampling distribution of for any value of from the alternative hypothesis, and the from the alternative hypothesis, and the acceptance region from step 4, compute the acceptance region from step 4, compute the probability that the sample mean will be in the probability that the sample mean will be in the acceptance region.acceptance region.

xx

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Determining the Sample SizeDetermining the Sample Sizefor a Hypothesis Test About a Population for a Hypothesis Test About a Population

MeanMean

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

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

nz z

a

( )

( )

2 2

02

nz z

a

( )

( )

2 2

02

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The End of Chapter 9The End of Chapter 9

1 1 slide statistics for business and economics seventh edition andersonsweeneywilliams slides...

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