Trials in an experiment with a polygraph include 98 results that include 22 cases of wrong results and 76 cases of correct results. Use a 0.01 significance level to test the claim that such polygraph results are correct less than $80 \%$ of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method. Use the normal distribution as an approximation of the binomial distribution. c. $\mathrm{H}_{0}: \mathrm{p}=0.20$ $H_{1}: p \neq 0.20$ E. $H_{0}: p=0.80$ $H_{1}: p \neq 0.80$ D. $H_{0}: p=0.80$ $H_{1}: p<0.80$ F. $H_{0}: p=0.20$ $H_{1}: p<0.20$ The test statistic is $z=$ (Round to two decimal places as needed.) The P-value is (Round to four decimal places as needed.) Identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim. $H_{0}$. There sufficient evidence to support the claim that the polygraph results are correct less than $80 \%$ of the time.

Step 1 ：Identify the null hypothesis and the alternative hypothesis. The null hypothesis is that the polygraph results are correct 80% of the time, denoted as \(H_{0}: p=0.80\). The alternative hypothesis is that the polygraph results are correct less than 80% of the time, denoted as \(H_{1}: p<0.80\).

Step 2 ：Calculate the test statistic using the formula 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 proportion, and \(n\) is the sample size. In this case, \(n = 98\), \(x = 76\), and \(p_{hat} = 0.7755102040816326\), so the test statistic is approximately \(z=-0.61\).

Step 3 ：Calculate the P-value, which is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. Using the normal distribution, the P-value is approximately \(0.27\).

Step 4 ：Make a conclusion about the null hypothesis. Since the P-value is greater than the significance level of 0.01, we fail to reject the null hypothesis. This means that there is not enough evidence to support the claim that the polygraph results are correct less than 80% of the time.

Step 5 ：\(\boxed{\text{Final Answer: The null hypothesis is } H_{0}: p=0.80 \text{ and the alternative hypothesis is } H_{1}: p<0.80. \text{ The test statistic is approximately } z=-0.61 \text{ and the P-value is approximately } 0.27. \text{ There is not enough evidence to reject the null hypothesis, so we cannot support the claim that the polygraph results are correct less than } 80 \% \text{ of the time.}}\)