Question

trials in an experiment with a polygraph include 96 results that include 24 cases of wrong results and 72 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.
let p be the population proportion of correct polygraph results. identify the null and alternative hypotheses. choose the correct answer below
a. ( h_0:p = 0.80 )
( h_1:plt0.80 )
b. ( h_0:p = 0.20 )
( h_1:plt0.20 )
c. ( h_0:p = 0.80 )
( h_1:p
eq0.80 )
d. ( h_0:p = 0.20 )
( h_1:p
eq0.20 )
e. ( h_0:p = 0.20 )
( h_1:pgt0.20 )
f. ( h_0:p = 0.80 )
( h_1:pgt0.80 )
the test statistic is ( z=-1.22 ) (round to two decimal places as needed.)
the p - value is 111 (round to three decimal places as needed.)
identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim
( h_0 ). there sufficient evidence to support the claim that the polygraph results are correct less than 80% of the time.

Explanation:

Step1: Determine the null and alternative hypotheses

The claim is that polygraph results are correct less than 80% of the time. So the null hypothesis \(H_{0}:p = 0.80\) (the proportion of correct results is 80%) and the alternative hypothesis \(H_{1}:p<0.80\) (the proportion of correct results is less than 80%). So option A is correct.

Step2: Analyze the P - value and null hypothesis

Given \(z=-1.22\). For a one - tailed test (since \(H_{1}:p < 0.80\)), the P - value is the probability that \(Z<-1.22\). Using the standard normal table, \(P(Z < - 1.22)=0.111\).
Since the significance level \(\alpha = 0.05\) and \(P - value=0.111>0.05\), we fail to reject the null hypothesis.

Answer:

Fail to reject \(H_{0}\). There is not sufficient evidence to support the claim that the polygraph results are correct less than 80% of the time.