Save In a program designed to help patients stop smoking, 216 patients were given sustained care, and $81.5 \%$ of them were no longer smoking after one month. Use a 0.01 significance level to test the claim that $80 \%$ of patients stop smoking when given sustained care. Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. $\mathrm{H}_{0}: \mathrm{p} \neq 0.8$ \[ H_{1}: p=0.8 \] B \[ \begin{array}{l} H_{0}: p=0.8 \\ H_{1}: p \neq 0.8 \end{array} \] C. \[ \begin{array}{l} H_{0}: p=0.8 \\ H_{1}: p>0.8 \end{array} \] D. \[ \begin{array}{l} H_{0}: p=0.8 \\ H_{1}: p<0.8 \end{array} \] Identify the test statistic for this hypothesis test. The test statistic for this hypothesis test is (Round to two decimal places as needed.)

Step 1 ：Identify the null and alternative hypotheses for this test. The null hypothesis is usually a statement of no effect or no difference. In this case, the null hypothesis would be that the proportion of patients who stop smoking when given sustained care is 80%, or \(p=0.8\). The alternative hypothesis is what we are testing against the null hypothesis. Since the question is asking us to test the claim that more than 80% of patients stop smoking when given sustained care, the alternative hypothesis would be that the proportion of patients who stop smoking when given sustained care is greater than 80%, or \(p>0.8\).

Step 2 ：Identify the test statistic for this hypothesis test. The test statistic is a measure of how far our sample statistic (the proportion of patients in our sample who stopped smoking) is from the null hypothesis value (the claimed proportion of 80%). We can calculate this using the formula for the test statistic for a proportion, which is \((p_{hat} - p_0) / \sqrt{(p_0 * (1 - p_0)) / n}\), where \(p_{hat}\) is the sample proportion, \(p_0\) is the null hypothesis value, and \(n\) is the sample size.

Step 3 ：Given that \(n = 216\), \(p_{hat} = 0.815\), and \(p_0 = 0.8\), we can substitute these values into the formula to get the test statistic.

Step 4 ：The test statistic for this hypothesis test is \(\boxed{0.55}\).