Question

trials in an experiment with a polygraph include 98 results that include 22 cases of wrong results and 76 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:p>0.80 )
b. ( h_0:p = 0.20 )
( h_1:p<0.20 )
c. ( h_0:p = 0.80 )
( h_1:p<0.80 )
d. ( h_0:p = 0.80 )
( h_1:p
eq0.80 )
e. ( h_0:p = 0.20 )
( h_1:p>0.20 )
f. ( h_0:p = 0.20 )
( h_1:p
eq0.20 )
the test statistic is ( z=square ) (round to two decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 76\) (number of correct results) and \(n=98\). So \(\hat{p}=\frac{76}{98}\approx0.776\)

Step2: Calculate the test - statistic

The formula for the test - statistic \(z\) in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)

Given \(p = 0.80\) (from the null hypothesis \(H_0:p = 0.80\)), \(n = 98\), and \(\hat{p}\approx0.776\)

$$
LATEXBLOCK0
$$

Answer:

The test statistic \(z\approx - 0.60\)