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.