Problem
Conduct the following test at the $\alpha=0.01$ level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the P-value. Assume that the samples were obtained independently using simple random sampling. Test whether $p_{1} \neq p_{2}$. Sample data are $x_{1}=28, n_{1}=255, x_{2}=38$, and $n_{2}=302$. (a) Determine the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0}: \mathrm{p}_{1}=\mathrm{p}_{2}$ versus $\mathrm{H}_{1}: \mathrm{p}_{1}<\mathrm{p}_{2}$ C. $H_{0}: p_{1}=p_{2}$ versus $H_{1}: p_{1} \neq p_{2}$ B. $H_{0}: p_{1}=p_{2}$ versus $H_{1}: p_{1}>p_{2}$ D. $H_{0}: p_{1}=0$ versus $H_{1}: p_{1}=0$ Clear all Check answer
Solution
Step 1 ：The null hypothesis is always a statement of no effect or no difference. In this case, the null hypothesis would be that the proportions are equal, i.e., \(p_{1} = p_{2}\).
Step 2 ：The alternative hypothesis is what we are testing for. In this case, we are testing for a difference in proportions, so the alternative hypothesis would be \(p_{1} \neq p_{2}\).
Step 3 ：Therefore, the correct answer is C. \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1} \neq p_{2}\).
Step 4 ：Final Answer: \(\boxed{C. H_{0}: p_{1}=p_{2} \text{ versus } H_{1}: p_{1} \neq p_{2}}\)
