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Chapter 9: Problem 21

For each of the following examples of tests of hypotheses about \(\mu\), show the rejection and nonrejection regions on the sampling distribution of the sample mean assuming it is normal. a. A two-tailed test with \(\alpha=.01\) and \(n=100\) b. A left-tailed test with \(\alpha=.005\) and \(n=27\) c. A right-tailed test with \(\alpha=.025\) and \(n=36\)

Short Answer

Expert verified
For a two-tailed test, the rejection regions are Z \(\leq -2.576\) or Z \(\geq 2.576\), and the nonrejection region is when -2.576 \(\leq\) Z \(\leq\) 2.576. For a left-tailed test, the rejection region is when Z \(\leq -2.578\), and the nonrejection region is when Z > -2.578. For a right-tailed test, the rejection region is when Z \(\geq 1.960\), and the nonrejection region is when Z < 1.960.

Step by step solution

01

Solve test a - Two-tailed test

A two-tailed test divides the significance level across both tails of the distribution. So, for a \(\alpha = .01\), the significance level for each tail would be \(.01/2 = .005\). For \(\alpha/2 = .005\), the Z score from the Z-table is 卤2.576. Hence, the rejection region is when Z \(\leq -2.576\) or Z \(\geq 2.576\), and the nonrejection region is when -2.576 \(\leq\) Z \(\leq\) 2.576.
02

Solve test b - Left-tailed test

For a left-tailed test, all the significance level rests in the left tail of the distribution. For \(\alpha = .005\), the Z score is -2.578. Therefore, the rejection region is when Z \(\leq -2.578\), and the nonrejection region is when Z > -2.578.
03

Solve test c - Right-tailed test

For a right-tailed test, the entire significance level is located in the right tail of the distribution. For \(\alpha = .025\), the Z score is 1.960. Hence, the rejection region is when Z \(\geq 1.960\), and the nonrejection region is when Z < 1.960.

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Most popular questions from this chapter

The customers at a bank complained about long lines and the time they had to spend waiting for service. It is known that the customers at this bank had to wait 8 minutes, on average, before being served. The management made some changes to reduce the waiting time for its customers. A sample of 60 customers taken after these changes were made produced a mean waiting time of \(7.5\) minutes with a standard deviation of \(2.1\) minutes. Using this sample mean, the bank manager displayed a huge banner inside the bank mentioning that the mean waiting time for customers has been reduced by new changes. Do you think the bank manager's claim is justifiable? Use the \(2.5 \%\) significance level to answer this question. Use both approaches.

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Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.45, \quad H_{1}: p \neq .45, \quad n=100, \quad \hat{p}=.49, \quad \alpha=.10\) b. \(H_{0}: p=.72, \quad H_{1}: p<.72, \quad n=700, \quad \hat{p}=.64, \quad \alpha=.05\) c. \(H_{0}: p=.30, \quad H_{1}: p>.30, \quad n=200, \quad \hat{p}=.33, \quad \alpha=.01\)

In a 2009 nonscientific poll on www.ESPN.com, \(67 \%\) of the respondents believed that Roger Federer was going to defeat Andy Roddick in the 2009 Wimbledon Gentlemen's singles championship. Suppose that a survey of 150 tennis fans conducted in Europe at the same time resulted in 118 who believed that Federer was going to win. Perform a hypothesis test to determine if it is reasonable to conclude that the percentage of all European tennis fans who believed that Federer was going to win the 2009 championship was higher than \(67 \%\), the result in the ESPN.com poll. Use a \(2 \%\) significance level and both the \(p\) -value and the critical-value approaches.

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