Trials in an experiment with a polygraph include 99 results that include 22 cases of wrong results and 77 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 $\mathrm{p}$ 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.
\[
\begin{array}{l}
H_{0}: p=0.80 \\
H_{1}: p> 0.80
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: p=0.80 \\
H_{1}: p< 0.80
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p=0.80 \\
H_{1}: p \neq 0.80
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p=0.20 \\
H_{1}: p \neq 0.20
\end{array}
\]
F.
\[
\begin{array}{l}
H_{0}: p=0.20 \\
H_{1}: p> 0.20
\end{array}
\]
The test statistic is $z=\square$. (Round to two decimal places as needed.)
The P-value is $\square$. (Round to four decimal places as needed.)
Identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim. $\mathrm{H}_{0}$. There sufficient evidence to support the claim that the polygraph results are correct less than $80 \%$ of the time.
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Answer
There is not enough evidence to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the polygraph results are correct less than 80% of the time.
Step 1 ：Given values are: total number of trials \(n = 99\), number of correct results \(x = 77\), null hypothesis population proportion \(p0 = 0.80\), and significance level \(\alpha = 0.01\).
Step 2 ：Calculate the sample proportion \(p_{hat} = \frac{x}{n} = \frac{77}{99} = 0.7777777777777778\).
Step 3 ：Calculate the standard error \(se = \sqrt{\frac{p0 * (1 - p0)}{n}} = \sqrt{\frac{0.8 * (1 - 0.8)}{99}} = 0.04020151261036848\).
Step 4 ：Calculate the test statistic \(z = \frac{p_{hat} - p0}{se} = \frac{0.7777777777777778 - 0.8}{0.04020151261036848} = -0.5527707983925675\).
Step 5 ：Calculate the P-value using the cumulative distribution function of the normal distribution. The P-value is approximately 0.29.
Step 6 ：Since the P-value is greater than the significance level of 0.01, we fail to reject the null hypothesis.
Step 7 ：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 8 ：The null and alternative hypotheses are: \(H_{0}: p=0.80\) and \(H_{1}: p<0.80\).
Step 9 ：The test statistic is \(z=-0.55\).
Step 10 ：The P-value is \(0.29\).
Step 11 ：There is not enough evidence to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the polygraph results are correct less than 80% of the time.
