91Ó°ÊÓ

Suppose the consequences of making a Type I error are severe. Would you choose the level of significance, \(\alpha,\) to equal \(0.01,0.05,\) or \(0.10 ?\) Why?

In Problems \(3-8,\) test the hypothesis, using (a) the classical approach and then (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{aligned}&H_{0}: p=0.3 \text { versus } H_{1}: p>0.3\\\&n=200 ; x=75 ; \alpha=0.05\end{aligned}$$

Determine the critical value for a two-tailed test of a population mean with \(\sigma\) unknown at the \(\alpha=0.05\) level of significance with 12 degrees of freedom.

Determine the critical value for a left-tailed test of a population mean with \(\sigma\) unknown at the \(\alpha=0.05\) level of significance with 19 degrees of freedom.

Conduct the appropriate test. A simple random sample of size \(n=15\) is drawn from a population that is normally distributed. The sample mean is found to be \(23.8,\) and the sample standard deviation is found to be \(6.3 .\) Test whether the population mean is different from 25 at the \(\alpha=0.01\) level of significance.

Suppose that we are testing the hypotheses \(H_{0}: \mu=\mu_{0}\) versus \(H_{1}: \mu \neq \mu_{0}\) and we find the \(P\) -value to be 0.02 Explain what this means. Would you reject \(H_{0} ?\) Why?

Lipitor The drug Lipitor is meant to reduce total cholesterol and LDL cholesterol. In clinical trials, 19 out of 863 patients taking \(10 \mathrm{mg}\) of Lipitor daily complained of flulike symptoms. Suppose that it is known that \(1.9 \%\) of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than \(1.9 \%\) of Lipitor users experience flulike symptoms as a side effect at the \(\alpha=0.01\) level of signifi-cance?

To test \(H_{0}: \mu=20\) versus \(H_{1}: \mu<20,\) a simple random sample of size \(n=18\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=18.3\) and \(s=4.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a simple random sample of size \(n=13\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=4.9\) and \(s=1.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?

To test \(H_{0}: \mu=45\) versus \(H_{1}: \mu \neq 45,\) a simple random sample of size \(n=40\) is obtained. (a) Does the population have to be normally distributed to test this hypothesis by using the methods presented in this section? (b) If \(\bar{x}=48.3\) and \(s=8.5,\) compute the test statistic. (c) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (d) Determine and interpret the \(P\) -value. (e) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, will the researcher reject the null hypothesis? Why?