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. $H_{0}: p=0.20$
\[
H_{1}: P< 0.20
\]
Answer
Final Answer: \(\boxed{D}\)
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: \(\boxed{D}\)
