Trials in an experiment with a polygraph include 98 results that include 24 cases of wrong results and 74 cases of correct results. Use a 0.05 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. The test statistic is $z=\square$. (Round to twhdecimal places as needed.)

Step 1 ：Identify the null hypothesis and the alternative hypothesis. The null hypothesis is that the polygraph results are correct 80% of the time, and the alternative hypothesis is that the polygraph results are correct less than 80% of the time.

Step 2 ：Calculate the test statistic using the formula \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and \(n\) is the sample size. In this case, \(\hat{p} = \frac{74}{98}\), \(p_0 = 0.8\), and \(n = 98\).

Step 3 ：Use a Z-table or a statistical software to find the P-value. If the P-value is less than the significance level (0.05), reject the null hypothesis.

Step 4 ：Make a conclusion about the null hypothesis and the original claim based on the P-value. The test statistic is approximately -1.11 and the P-value is approximately 0.133. Since the P-value is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the polygraph results are correct less than 80% of the time.

Step 5 ：Final Answer: The test statistic is \(\boxed{-1.11}\).