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Critical Value Approach Fill in the blanks in the table below. $$\begin{array}{|l|l|l|l|l|l|}\hline \begin{array}{l}\text { Test } \\\\\text { Statistic }\end{array} & \begin{array}{l}\text { Significance } \\\\\text { Level }\end{array} &\begin{array}{l}\text { One or } \\\\\text { Two-Tailed Test? }\end{array} & \text { Critical Value } & \begin{array}{l}\text { Rejection } \\\\\text { Region }\end{array} & \text { Conclusion } \\\\\hline z=0.88 & \alpha=.05 & \text { Two-tailed } & & & \\\\\hline z=-2.67 & \alpha=.05 & \text { 0ne-tailed (lower) } & & & \\\\\hline z=5.05 & \alpha=.01 & \text { Two-tailed } & & & \\\\\hline z=-1.22 & \alpha=.01 & \text { One- tailed (lower) } & & & \\\\\hline\end{array}$$

Find the appropriate rejection regions for the large-sample test statistic \(z\) in these cases: a. A right-tailed test with \(\alpha=.01\) b. A two-tailed test at the \(5 \%\) significance level c. A left-tailed test at the \(1 \%\) significance level d. A two-tailed test with \(\alpha=01\)

A random sample of \(n=35\) observations from a quantitative population produced a mean \(\bar{x}=2.4\) and a standard deviation \(s=.29 .\) Suppose your research objective is to show that the population mean \(\mu\) exceeds 2.3 a. Give the null and alternative hypotheses for the test. b. Locate the rejection region for the test using a \(5 \%\) significance level. c. Find the standard error of the mean. d. Before you conduct the test, use your intuition to decide whether the sample mean \(\bar{x}=2.4\) is likely or unlikely, assuming that \(\mu=2.3 .\) Now conduct the test. Do the data provide sufficient evidence to indicate that \(\mu>2.3 ?\)

Flextime Many companies are becoming involved in flextime, in which a worker schedules his or her own work hours or compresses work weeks. A company that was contemplating the installation of a flextime schedule estimated that it needed a minimum mean of 7 hours per day per assembly worker in order to operate effectively. Each of a random sample of 80 of the company's assemblers was asked to submit a tentative flextime schedule. If the mean number of hours per day for Monday was 6.7 hours and the standard deviation was 2.7 hours, do the data provide sufficient evidence to indicate that the mean number of hours worked per day on Mondays, for all of the company's assemblers, will be less than 7 hours? Test using \(\alpha=.05\)

What's Normal? What is normal, when it comes to people's body temperatures? A random sample of 130 human body temperatures, provided by Allen Shoemaker \(^{3}\) in the Journal of Statistical Education, had a mean of 98.25 degrees and a standard deviation of 0.73 degrees. Does the data indicate that the average body temperature for healthy humans is different from 98.6 degrees, the usual average temperature cited by physicians and others? Test using both methods given in this section. a. Use the \(p\) -value approach with \(\alpha=.05\). b. Use the critical value approach with \(\alpha=.05 .\) c. Compare the conclusions from parts a and b. Are they the same? d. The 98.6 standard was derived by a German doctor in \(1868,\) who claimed to have recorded 1 million temperatures in the course of his research. \({ }^{4}\) What conclusions can you draw about his research in light of your conclusions in parts a and \(b\) ?

Early Detection of Breast Cancer Of those women who are diagnosed to have early-stage breast cancer, one-third eventually die of the disease. Suppose a community public health department instituted a screening program to provide for the early detection of breast cancer and to increase the survival rate \(p\) of those diagnosed to have the disease. A random sample of 200 women was selected from among those who were periodically screened by the program and who were diagnosed to have the disease. Let \(x\) represent the number of those in the sample who survive the disease a. If you wish to detect whether the community screening program has been effective, state the null hypothesis that should be tested. b. State the alternative hypothesis. c. If 164 women in the sample of 200 survive the disease, can you conclude that the community screening program was effective? Test using \(\alpha=.05\) and explain the practical conclusions from your test. d. Find the \(p\) -value for the test and interpret it.

Treatment versus Control An experiment was conducted to test the effect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50\. The first group, the control group, received no treatment for the infection. The second group received the drug. After a 30 -day period, the proportions of survivors, \(\hat{p}_{1}\) and \(\hat{p}_{2}\), in the two groups were found to be .36 and \(.60,\) respectively. a. Is there sufficient evidence to indicate that the drug is effective in treating the viral infection? Use \(\alpha=.05 .\) b. Use a \(95 \%\) confidence interval to estimate the actual difference in the cure rates for the treated versus the control groups.

a. Define \(\alpha\) and \(\beta\) for a statistical test of hypothesis. b. For a fixed sample size \(n\), if the value of \(\alpha\) is decreased, what is the effect on \(\beta\) ? c. In order to decrease both \(\alpha\) and \(\beta\) for a particular alternative value of \(\mu,\) how must the sample size change?

Daily Wages The daily wages in a particular industry are normally distributed with a mean of \(\$ 94\) and a standard deviation of \(\$ 11.88\). Suppose a company in this industry employs 40 workers and pays them \(\$ 91.50\) per week on the average. Can these workers be viewed as a random sample from among all workers in the industry? a. What are the null and alternative hypotheses to be tested? b. Use the Large-Sample Test of a Population Mean applet to find the observed value of the test statistic. c. Use the Large-Sample Test of a Population Mean applet to find the \(p\) -value for this test. d. If you planned to conduct your test using \(\alpha=.01\), what would be your test conclusions? e. Was it necessary to know that the daily wages are normally distributed? Explain your answer.