One-Sample Hypothesis Testing

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc.

One-Sample Hypothesis One-Sample Hypothesis Testing Testing

Chapter9999

Logic of Hypothesis Testing

Statistical Hypothesis Testing

Testing a Mean: Known Population Variance

Testing a Mean: Unknown Population Variance

Testing a Proportion

Power Curves and OC Curves (Optional)

Tests for One Variance (Optional)

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Logic of Hypothesis TestingLogic of Hypothesis Testing

• Steps in Hypothesis TestingSteps in Hypothesis Testing

Step 1: State the assumption to be testedStep 1: State the assumption to be tested

Step 2: Specify the Step 2: Specify the decision ruledecision rule

Step 3: Collect the data to test the Step 3: Collect the data to test the hypothesis hypothesis

Step 4: Make a decisionStep 4: Make a decision

Step 5: Take action based on the decisionStep 5: Take action based on the decision

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• Hypotheses are a pair of mutually Hypotheses are a pair of mutually exclusive, collectively exhaustive exclusive, collectively exhaustive statements about the world.statements about the world.

• One statement or the other must be true, One statement or the other must be true, but they cannot both be true.but they cannot both be true.

• HH00: Null Hypothesis: Null Hypothesis

HH11: Alternative Hypothesis: Alternative Hypothesis

• These two statements are hypotheses These two statements are hypotheses because the truth is unknown.because the truth is unknown.

State the HypothesisState the Hypothesis

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• Efforts will be made to reject the null Efforts will be made to reject the null hypothesis.hypothesis.

• If If HH00 is rejected, we tentatively conclude is rejected, we tentatively conclude HH11

to be the case. to be the case.

• HH00 is sometimes called the is sometimes called the maintained maintained

hypothesis.hypothesis.

• HH11 is called the is called the action alternativeaction alternative because because

action may be required if we reject action may be required if we reject HH00 in in

favor of favor of HH11..

State the HypothesisState the Hypothesis

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• We cannot prove a null hypothesis, we can We cannot prove a null hypothesis, we can only only fail to rejectfail to reject it. it.

Can Hypotheses be Proved?Can Hypotheses be Proved?

• The null hypothesis is The null hypothesis is assumed trueassumed true and a and a contradiction is sought.contradiction is sought.

Role of EvidenceRole of Evidence

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Types of ErrorTypes of Error• Type I errorType I error: Rejecting the null hypothesis : Rejecting the null hypothesis

when it is true. This occurs with probability when it is true. This occurs with probability ..

• Type II errorType II error: Failure to reject the null : Failure to reject the null hypothesis when it is false. This occurs hypothesis when it is false. This occurs with probability with probability ..

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Statistical Hypothesis TestingStatistical Hypothesis Testing

• A A statistical hypothesisstatistical hypothesis is a statement about is a statement about the value of a population parameter the value of a population parameter ..

• A A hypothesis testhypothesis test is a decision between two is a decision between two competing mutually exclusive and competing mutually exclusive and collectively exhaustive hypotheses about collectively exhaustive hypotheses about the value of the value of ..

Left-Tailed Test Right-Tailed TestTwo-Tailed Test

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• The direction of the test is indicated by The direction of the test is indicated by HH11::

> indicates a right-tailed test< indicates a left-tailed test≠ indicates a two-tailed test

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When to use a One- or Two-Sided TestWhen to use a One- or Two-Sided Test• A two-sided hypothesis test (i.e., A two-sided hypothesis test (i.e., ≠ ≠ 00) is used ) is used

when direction (< or >) is of no interest to the when direction (< or >) is of no interest to the decision maker decision maker

• A one-sided hypothesis test is used when A one-sided hypothesis test is used when - the consequences of rejecting - the consequences of rejecting HH00 are are

asymmetric, or asymmetric, or - where one tail of the distribution is of special - where one tail of the distribution is of special

importance to the researcher. importance to the researcher.• Rejection in a two-sided test guarantees Rejection in a two-sided test guarantees

rejection in a one-sided test, other things rejection in a one-sided test, other things being equal.being equal.

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Decision RuleDecision Rule• A A test statistictest statistic shows how far the sample shows how far the sample

estimate is from its expected value, in estimate is from its expected value, in terms of its own standard error.terms of its own standard error.

• The The decision ruledecision rule uses the known sampling uses the known sampling distribution of the test statistic to establish distribution of the test statistic to establish the the critical valuecritical value that divides the sampling that divides the sampling distribution into two regions.distribution into two regions.

• Reject Reject HH00 if the test statistic lies in the if the test statistic lies in the

rejection regionrejection region..

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Decision Rule for Two-Tailed TestDecision Rule for Two-Tailed Test• Reject Reject HH00 if the test statistic < left-tail if the test statistic < left-tail

critical value or if the test statistic > right-critical value or if the test statistic > right-tail critical value.tail critical value.

Figure 9.2

- Critical value + Critical value

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Decision Rule for Left-Tailed TestDecision Rule for Left-Tailed Test• Reject Reject HH00 if the test statistic < left-tail if the test statistic < left-tail

critical value.critical value.

Figure 9.2

- Critical value

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Decision Rule for Right-Tailed TestDecision Rule for Right-Tailed Test• Reject Reject HH00 if the test statistic > right-tail if the test statistic > right-tail

critical value.critical value.

+ Critical value

Figure 9.2

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Type I ErrorType I Error• , the probability of a Type I error, is the , the probability of a Type I error, is the level level

of significanceof significance (i.e., the probability that the test (i.e., the probability that the test statistic falls in the rejection region even statistic falls in the rejection region even though though HH00 is true). is true).

= = PP(reject (reject HH00 | | HH00 is true) is true)

• A Type I error is sometimes referred to as a A Type I error is sometimes referred to as a false positive.false positive.

• For example, if we choose For example, if we choose = .05, we expect to = .05, we expect to commit a Type I error about 5 times in 100.commit a Type I error about 5 times in 100.

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Type I ErrorType I Error• A small A small is desirable, other things being is desirable, other things being

equal.equal.• Chosen in advance, common choices for Chosen in advance, common choices for are are

.10, .05, .025, .01 and .005 (i.e., 10%, 5%, 2.5%, 1% and .5%).

• The The risk is the area under the tail(s) of the risk is the area under the tail(s) of the sampling distribution.sampling distribution.

• In a two-sided test, the In a two-sided test, the risk is split with risk is split with /2 in /2 in each tail since there are two ways to reject each tail since there are two ways to reject HH00..

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Type II ErrorType II Error• , the probability of a type II error, is the , the probability of a type II error, is the

probability that the test statistic falls in the probability that the test statistic falls in the acceptance region even though acceptance region even though HH00 is false. is false.

= = PP(fail to reject (fail to reject HH00 | | HH00 is false) is false)

• cannot be chosen in advance because it cannot be chosen in advance because it depends on depends on and the sample size. and the sample size.

• A small A small is desirable, other things being is desirable, other things being equal.equal.

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Power of a TestPower of a Test• The The powerpower of a test is the probability that a of a test is the probability that a

false hypothesis will be rejected.false hypothesis will be rejected.• Power = 1 – Power = 1 – • A low A low risk means high power. risk means high power.

• Larger samples lead to increased power.Larger samples lead to increased power.

Power = Power = PP(reject (reject HH00 | | HH00 is false) = 1 – is false) = 1 –

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Relationship Between Relationship Between and and • Both a small Both a small and a small and a small are desirable. are desirable.• For a given type of test and fixed sample For a given type of test and fixed sample

size, there is a trade-off between size, there is a trade-off between and and ..• The larger critical value needed to reduce The larger critical value needed to reduce

risk makes it harder to reject risk makes it harder to reject HH00, thereby , thereby

increasing increasing risk. risk.• Both Both and and can be reduced can be reduced

simultaneously only by increasing the simultaneously only by increasing the sample size.sample size.

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Consequences of a Type II ErrorConsequences of a Type II Error• Firms are increasingly wary of Type II errror Firms are increasingly wary of Type II errror

(failing to recall a product as soon as sample (failing to recall a product as soon as sample evidence begins to indicate potential problems.)evidence begins to indicate potential problems.)

Significance versus ImportanceSignificance versus Importance• The standard error of most sample estimators The standard error of most sample estimators

approaches 0 as sample size increases. approaches 0 as sample size increases.

• In this case, no matter how small, In this case, no matter how small, – – 00 will be will be

significant if the sample size is large enough.significant if the sample size is large enough.• Therefore, expect Therefore, expect significantsignificant effects even when an effects even when an

effect is too slight to have any effect is too slight to have any practical practical importanceimportance..

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Testing a Mean: Testing a Mean: Known Population VarianceKnown Population Variance

• The hypothesized mean The hypothesized mean 00 that we are that we are

testing is a benchmark.testing is a benchmark.

• The value of The value of 00 does not come from a does not come from a

sample.sample.• The The test statistictest statistic compares the sample mean compares the sample mean

xx with the hypothesized mean with the hypothesized mean 00..

• The difference between The difference between xx and and 00 is divided by is divided by

the standard error of the mean (denoted the standard error of the mean (denoted xx).).

• The test statistic isThe test statistic is

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Testing the HypothesisTesting the Hypothesis• Step 1: State the hypothesesStep 1: State the hypotheses

For example, For example, HH00: : << 216 216 mmmm

HH11: : > 216 > 216 mmmm

• Step 2: Specify the decision ruleStep 2: Specify the decision ruleFor example, for For example, for = .05 = .05 for the right-tail area, for the right-tail area, Reject Reject HH00 if if zz > 1.645, > 1.645,

otherwise do not otherwise do not

reject reject HH00

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Testing the HypothesisTesting the Hypothesis• For a two-tailed test, we split the risk of For a two-tailed test, we split the risk of

Type I error by putting Type I error by putting /2 in each tail. /2 in each tail. For example, for For example, for = .05 = .05

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Testing the HypothesisTesting the Hypothesis• Step 3: Calculate the test statisticStep 3: Calculate the test statistic

• Step 4: Make the decisionStep 4: Make the decisionIf the test statistic falls in the rejection If the test statistic falls in the rejection region as defined by the critical value, we region as defined by the critical value, we reject reject HH00 and conclude and conclude HH11..

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Analogy to Confidence IntervalsAnalogy to Confidence Intervals

• A two-tailed hypothesis test at the 5% level A two-tailed hypothesis test at the 5% level of significance (of significance ( = .05) is exactly = .05) is exactly equivalent to asking whether the 95% equivalent to asking whether the 95% confidence interval for the mean includes confidence interval for the mean includes the hypothesized mean.the hypothesized mean.

• If the confidence interval includes the If the confidence interval includes the hypothesized mean, then we cannot reject hypothesized mean, then we cannot reject the null hypothesis.the null hypothesis.

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Using the p-Value ApproachUsing the p-Value Approach• The The pp-value is the probability of the sample -value is the probability of the sample

result (or one more extreme) assuming that result (or one more extreme) assuming that HH00 is true. is true.

• The The pp-value can be obtained using Excel’s -value can be obtained using Excel’s cumulative standard normal functioncumulative standard normal function=NORMSDIST(=NORMSDIST(zz))

• The The pp-value can also be obtained from -value can also be obtained from Appendix C-2.Appendix C-2.

• Using the Using the pp-value, we reject -value, we reject HH00 if if pp-value -value << ..

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Testing a Mean: Testing a Mean: Unknown Population VarianceUnknown Population Variance

• When the population standard deviation When the population standard deviation is unknown and the population may be is unknown and the population may be assumed normal, the test statistic follows assumed normal, the test statistic follows the Student’s the Student’s tt distribution with distribution with = = nn – 1 – 1 degrees of freedom.degrees of freedom.

• The test statistic isThe test statistic is

Using Student’s tUsing Student’s t

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Testing a Mean: Testing a Mean: Unknown Population Variance Unknown Population Variance

Testing a HypothesisTesting a Hypothesis• Step 1: State the hypothesesStep 1: State the hypotheses

For example, For example, HH00: : = 142 = 142

HH11: : ≠ 142 ≠ 142

• Step 2: Specify the decision ruleStep 2: Specify the decision ruleFor example, for For example, for = .10 = .10 for a two-tailed area, for a two-tailed area, Reject Reject HH00 if if tt > 1.714 or > 1.714 or

tt < -1.714, otherwise < -1.714, otherwise do not reject do not reject HH00

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Testing a Mean: Testing a Mean: Unknown Population Variance Unknown Population Variance

Testing a HypothesisTesting a Hypothesis• Step 3: Calculate the test statisticStep 3: Calculate the test statistic

• Step 4: Make the decisionStep 4: Make the decisionIf the test statistic falls in the rejection If the test statistic falls in the rejection region as defined by the critical values, we region as defined by the critical values, we reject reject HH00 and conclude and conclude HH11..

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Testing a Mean: Testing a Mean: Unknown Population VarianceUnknown Population Variance

Confidence Intervals versus Hypothesis TestConfidence Intervals versus Hypothesis Test• A two-tailed hypothesis test at the 10% A two-tailed hypothesis test at the 10%

level of significance (level of significance ( = .10) is equivalent = .10) is equivalent to a two-sided 90% confidence interval for to a two-sided 90% confidence interval for the mean.the mean.

• If the confidence interval does not include If the confidence interval does not include the hypothesized mean, then we reject the the hypothesized mean, then we reject the null hypothesis.null hypothesis.

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Testing a ProportionTesting a Proportion

• To conduct a hypothesis test, we need to knowTo conduct a hypothesis test, we need to know- the parameter being tested- the parameter being tested- the sample statistic- the sample statistic- the sampling distribution of the sample - the sampling distribution of the sample statisticstatistic

• The sampling distribution tells us which test The sampling distribution tells us which test statistic to use.statistic to use.

• A sample proportion A sample proportion pp estimates the estimates the population proportion population proportion ..

• Remember that for a large sample, Remember that for a large sample, p p can be can be assumed to follow a normal distribution. If assumed to follow a normal distribution. If so, the test statistic is so, the test statistic is zz..

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pp = = xxnn

== number of successesnumber of successessample sizesample size

zzcalccalc = = pp – – 00

pp

• If If nn00 >> 10 and 10 and nn(1-(1-00) ) >> 10, then 10, then

Where Where pp = = 00(1-(1-00))

nn

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• The value of The value of 00 that we are testing is a that we are testing is a

benchmarkbenchmark such as past experience, an such as past experience, an industry standard, or a product industry standard, or a product specification. specification.

• The value of The value of 00 does not come from a does not come from a

sample.sample.

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Critical ValueCritical Value• The test statistic is compared with a The test statistic is compared with a critical critical

valuevalue from a table. from a table.• The critical value shows the range of The critical value shows the range of

values for the test statistic that would be values for the test statistic that would be expected by chance if the expected by chance if the HH00 were true. were true.

Level of Sig. (Level of Sig. ()) Two-tailed Two-tailed testtest

Right-Tailed Right-Tailed TestTest

Left-Tailed Left-Tailed TestTest

.10.10 ++ 1.645 1.645 1.2821.282 -1.282-1.282

.05.05 ++1.9601.960 1.6451.645 -1.645-1.645

.01.01 ++ 2.576 2.576 2.3262.326 -2.326-2.326

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Steps in Testing a ProportionSteps in Testing a Proportion• Step 1: State the hypothesesStep 1: State the hypotheses

For example, For example, HH00: : >> .13 .13

HH11: : < .13 < .13

• Step 2: Specify the decision ruleStep 2: Specify the decision ruleFor example, for For example, for = .05 = .05for a left-tail area, for a left-tail area, reject reject HH00 if if zz < -1.645, < -1.645,

otherwise do not otherwise do not

reject reject HH00Figure 9.12

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Steps in Testing a ProportionSteps in Testing a Proportion• For a two-tailed test, we split the risk of For a two-tailed test, we split the risk of

type I error by putting type I error by putting /2 in each tail. /2 in each tail. For example, for For example, for = .05 = .05

Figure 9.14

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Steps in Testing a ProportionSteps in Testing a Proportion• Now, check the normality assumption: Now, check the normality assumption:

nn00 >> 10 and 10 and nn(1-(1-00) ) >> 10. 10.

• Step 3: Calculate the test statisticStep 3: Calculate the test statistic

• Step 4: Make the decisionStep 4: Make the decisionIf the test statistic falls in the rejection If the test statistic falls in the rejection region as defined by the critical value, we region as defined by the critical value, we reject reject HH00 and conclude and conclude HH11..

zz = = pp – – 00

pp

Where Where pp = = 00(1-(1-00))

nn

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Using the p-ValueUsing the p-Value• The The pp-value is the probability of the sample -value is the probability of the sample

result (or one more extreme) assuming that result (or one more extreme) assuming that HH00 is true. is true.

• The The pp-value can be obtained using Excel’s -value can be obtained using Excel’s cumulative standard normal functioncumulative standard normal function=NORMSDIST(=NORMSDIST(zz))

• The The pp-value can also be obtained from -value can also be obtained from Appendix C-2.Appendix C-2.

• Using the Using the pp-value, we reject -value, we reject HH00 if if pp-value -value << ..

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Using the p-ValueUsing the p-Value

• The The pp-value is a direct measure of the level -value is a direct measure of the level of significance at which we could reject of significance at which we could reject HH00. .

• Therefore, the Therefore, the smallersmaller the the pp-value, the more -value, the more we want to we want to rejectreject HH00..

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Calculating a p-Value for a Two-Tailed TestCalculating a p-Value for a Two-Tailed Test• For a two-tailed test, we divide the risk into For a two-tailed test, we divide the risk into

equal tails. So, to compare the equal tails. So, to compare the pp-value to -value to , first combine the , first combine the pp-values in the two tail -values in the two tail areas.areas.

• For example, if our test statistic was -1.975, For example, if our test statistic was -1.975, thenthen

2 x 2 x PP((zz < -1.975) = 2 x .02413 = .04826 < -1.975) = 2 x .02413 = .04826

At At = .05, we would reject = .05, we would reject HH00 since since

pp-value = .04826 < -value = .04826 < ..

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Effect of Effect of • No matter which level of significance you use, No matter which level of significance you use,

the test statistic the test statistic remains the remains the same. For same. For example, for a example, for a test statistic of test statistic of zz = 2.152 = 2.152

- 2.576- 2.576 2.5762.576- 1.645- 1.645 1.6451.645 z = 2.152z = 2.152

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Small Samples and Non-NormalitySmall Samples and Non-Normality

• In the case where In the case where nn00 < 10, use < 10, use MINITABMINITAB to to

test the hypotheses by finding the exact test the hypotheses by finding the exact binomial probability of a sample proportion binomial probability of a sample proportion pp. . For example,For example,

Figure 9.19

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Power Curves and OC Curves Power Curves and OC Curves (Optional)(Optional)

• Power depends on how far the true value of Power depends on how far the true value of the parameter is from the null hypothesis the parameter is from the null hypothesis value.value.

• The further away the true population value is The further away the true population value is from the assumed value, the easier it is for from the assumed value, the easier it is for your hypothesis test to detect and the more your hypothesis test to detect and the more power it has.power it has.

• Remember that Remember that

Power Curves for a MeanPower Curves for a Mean

= = PP(accept (accept HH00 | | HH00 is false) is false)

Power = Power = PP(reject (reject HH00 | | HH00 is false) = 1 – is false) = 1 –

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Power Curves and OC CurvesPower Curves and OC Curves

• We want power to be as close to 1 as possible.We want power to be as close to 1 as possible.• The values of The values of and power will vary, depending and power will vary, depending

on on - the difference between the true mean - the difference between the true mean and and thethe hypothesized mean hypothesized mean 00,,

- the standard deviation, - the standard deviation, - the sample size - the sample size nn and and - the level of significance - the level of significance

Power Curves for a MeanPower Curves for a Mean

Power = Power = ff(( – – 00, , , , nn, , ) )

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• We can get more power by increasing We can get more power by increasing , , but we would then increase the probability but we would then increase the probability of a type I error.of a type I error.

• A better way to increase power is to A better way to increase power is to increase the sample size increase the sample size nn..

Power Curves for a MeanPower Curves for a MeanTable 9.8

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Calculating PowerCalculating Power

• Step 1: Find the left-tail Step 1: Find the left-tail critical valuecritical value for for the sample mean.the sample mean.

• For any given values of For any given values of , , , , nn, and , and , and , and the assumption that the assumption that XX is normally is normally distributed, use the following steps to distributed, use the following steps to calculate calculate and power. and power.

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• The probability of The probability of error is the area to the error is the area to the right of the critical value right of the critical value xxcriticalcritical which which

represents represents PP((xx > > xxcriticalcritical | | = = 00))


Calculating PowerCalculating Power

• The decision rule is:The decision rule is:Reject Reject HH00 if if xx < < xxcriticalcritical

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Calculating PowerCalculating Power• Step 2: Express the difference between the Step 2: Express the difference between the

critical value critical value xxcriticalcritical and the true mean and the true mean as a as a

z-value:z-value:

• Step 3: Find the Step 3: Find the risk and power as areas risk and power as areas under the normal curve using Appendix C-2 under the normal curve using Appendix C-2 or Excel.or Excel. = = PP((xx > > xxcriticalcritical || = = 00))

Power = Power = PP((xx < < xxcriticalcritical | | = = 00) = 1 – ) = 1 –

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Effect of Sample SizeEffect of Sample Size• Other things being equal, if sample size Other things being equal, if sample size

were to increase, were to increase, risk would decline and risk would decline and power would increase because the critical power would increase because the critical value value xxcriticalcritical would be closer to the would be closer to the

hypothesized mean hypothesized mean ..

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• Here is a family of power curves.Here is a family of power curves.

Relationship of the Power and OC CurvesRelationship of the Power and OC Curves

Figure 9.22

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• The graph of The graph of risk against this same risk against this same XX-axis -axis is called the is called the operating characteristic operating characteristic or or OCOC curve.curve.

Relationship of the Power and OC CurvesRelationship of the Power and OC Curves

Figure 9.23

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Power Curve for Tests of a ProportionPower Curve for Tests of a Proportion

• Power depends on Power depends on - the true proportion (- the true proportion (),),- the hypothesized proportion (- the hypothesized proportion (00))

- the sample size (- the sample size (nn))- the level of significance (- the level of significance ())

Table 9.10

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Calculating PowerCalculating Power• Step 1: Find the left-tail Step 1: Find the left-tail critical valuecritical value for for

the sample proportion.the sample proportion.

• Step 2: Express the difference between the Step 2: Express the difference between the critical value critical value ppcriticalcritical and the true proportion and the true proportion

as a as a zz-value:-value:

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Power = Power = PP((pp > > ppcriticalcritical | | > > 00) = 1 – ) = 1 –


Calculating PowerCalculating Power• Step 3: Find the Step 3: Find the risk and power as areas risk and power as areas

under the normal curve.under the normal curve.

= = PP((pp < < ppcriticalcritical || = = 00))

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Using LearningStatsUsing LearningStats• Create power curves for a mean or Create power curves for a mean or

proportion without tedious calculations.proportion without tedious calculations.

Figure 9.25

9-55


Using Visual StatisticsUsing Visual Statistics• Create power curves for a mean or Create power curves for a mean or

proportion without tedious calculations.proportion without tedious calculations.

Figure 9.26

9-56Reject Reject HH00 if if 22 < < 22

lowerlower or if or if 22 > > 22upperupper

Tests for One VarianceTests for One Variance

• Sometimes we want to compare the variance Sometimes we want to compare the variance of a process with a historical benchmark or of a process with a historical benchmark or other standard.other standard.

• For a two-tailed test, the hypotheses are:For a two-tailed test, the hypotheses are:HH00: : 22 = = 22

0 0 HH11 = = 22 ≠ ≠ 2200

• For a test of one variance, assuming a normal For a test of one variance, assuming a normal population, the statistic population, the statistic 22 follows the follows the chi-chi-square distribution square distribution with degrees of freedom with degrees of freedom equal to equal to = n = n – – 1.1.

2 2 = = ((nn – 1) – 1)22

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Applied Statistics in Applied Statistics in Business & EconomicsBusiness & Economics

End of Chapter 9End of Chapter 9

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One-Sample Hypothesis Testing