One-Sample Hypothesis One-Sample Hypothesis TestsTests

Chapter9999

Logic of Hypothesis Testing

Statistical Hypothesis Testing

Testing a Mean: Known Population Variance

Testing a Mean: Unknown Population Variance

Testing a Proportion

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McGraw-Hill/Irwin

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 hypothesisStep 3: Collect the data to test the 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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Logic of Hypothesis TestingLogic of Hypothesis Testing

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

• HH00: Null Hypothesis: Null HypothesisHH11: Alternative Hypothesis: Alternative Hypothesis

• Efforts will be made to reject the null hypothesis. Efforts will be made to reject the null hypothesis. The null hypothesis is The null hypothesis is assumed trueassumed true and a and a contradiction is sought.contradiction is sought.

• If If HH00 is rejected, we tentatively conclude is rejected, we tentatively conclude HH11 to be to be the case. the case.

State the HypothesisState the Hypothesis

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Logic of Hypothesis TestingLogic of Hypothesis Testing

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

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

when it is false. This occurs with probability when it is false. This occurs 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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Statistical Hypothesis TestingStatistical Hypothesis Testing

• 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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Statistical Hypothesis TestingStatistical Hypothesis Testing

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

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

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

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

regionregion..

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Statistical Hypothesis TestingStatistical Hypothesis Testing

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

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

Figure 9.2

- Critical value + Critical value

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Statistical Hypothesis TestingStatistical Hypothesis Testing

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

Figure 9.2

- Critical value

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Statistical Hypothesis TestingStatistical Hypothesis Testing

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

value.value.

+ Critical value

Figure 9.2

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

• The The test statistictest statistic compares the sample mean compares the sample mean xx with with the hypothesized mean the hypothesized mean 00..

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

standard error of the mean (denoted standard error of the mean (denoted xx).).• The test statistic isThe test statistic is

0:1 0:1 0:1 0:0 0:0 0:0

HHH

HHH

led Test Right-Tai TestTwo-TailedTestTailedLeft

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Testing the HypothesisTesting the Hypothesis• Make the decision:Make the decision:

If the test statistic falls in the rejection region as If the test statistic falls in the rejection region as defined by the critical value, we reject defined by the critical value, we reject HH00 and and

conclude conclude HH11..

Using the p-Value ApproachUsing the p-Value Approach

• The The pp-value is the probability of the sample result -value is the probability of the sample result (or one more extreme) assuming that (or one more extreme) assuming that HH00 is true. is true.

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

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

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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

ttcalccalc = = xx – – 00

s/ ns/ n

Using Student’s tUsing Student’s t

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Testing a ProportionTesting a Proportion

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 the test statistic is 10, then the test statistic is

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

nn

zzcalccalc will be compared to a critical value depending on .

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Testing a ProportionTesting a Proportion

• Reject the null hypothesis if the test statistic falls Reject the null hypothesis if the test statistic falls

in the rejection region.in the rejection region.

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

NOTE:NOTE: 1. A two-tailed hypothesis test at the 5% level of 1. A two-tailed hypothesis test at the 5% level of significance (significance ( = .05) is exactly equivalent to asking whether = .05) is exactly equivalent to asking whether the 95% confidence interval for the mean (proportion) includes the 95% confidence interval for the mean (proportion) includes the hypothesized mean (proportion). 2. If the confidence the hypothesized mean (proportion). 2. If the confidence interval includes the hypothesized mean (proportion) , then we interval includes the hypothesized mean (proportion) , then we cannot reject the null hypothesis.cannot reject the null hypothesis.

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