Problem
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. Let $p$ be the population proportion of correct polygraph results. Identify the null and alternative hypotheses. Choose the correct answer below. A. \[ \begin{array}{l} H_{0}: p=0.20 \\ H_{1}: p>0.20 \end{array} \] c. \[ \begin{array}{l} H_{0}: p=0.20 \\ H_{1}: p \neq 0.20 \end{array} \] E. \[ \begin{array}{l} H_{0}: p=0.80 \\ H_{1}: p \neq 0.80 \end{array} \] B. $H_{0}: p=0.80$ \[ H_{1}: p>0.80 \] D. $H_{0}: p=0.80$ \[ H_{1}: p<0.80 \] F. $H_{0}: p=0.20$ \[ H_{1}: P<0.20 \]
Solution
Step 1 ：Identify the null and alternative hypotheses. 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 correct polygraph results is equal to 80%. The alternative hypothesis is what we are testing for. In this case, we are testing the claim that the proportion of correct polygraph results is less than 80%. Therefore, the null and alternative hypotheses are as follows: Null hypothesis, \(H_{0}: p=0.80\) Alternative hypothesis, \(H_{1}: p<0.80\)
Step 2 ：So, the correct answer is D. \(H_{0}: p=0.80\), \(H_{1}: p<0.80\)
Step 3 ：Final Answer: \(\boxed{D}\)
