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 (square). (round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{72}{96} = 0.75$. The null hypothesis is $H_{0}:p = 0.80$ and the alternative hypothesis is $H_{1}:p<0.80$ (as we are testing if the proportion of correct results is less than $80\%$).

Step2: Find the P - value

Since the test is left - tailed (because $H_{1}:p < 0.80$) and the test statistic $z=-1.22$.
We use the standard normal distribution table or a calculator with a normal - distribution function. The P - value for a left - tailed test with $z=-1.22$ is $P(Z < - 1.22)$.
Using a standard normal table or a calculator (e.g., in Excel: =NORM.S.DIST(-1.22,TRUE)), we get $P(Z < - 1.22)=0.111$

Answer:

The P - value is $0.111$