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=square ). (round to two decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 72\) (correct results) and \(n=96\). So \(\hat{p}=\frac{72}{96}=0.75\)

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 \(p = 0.8\) (from the null hypothesis \(H_0:p = 0.8\)), \(\hat{p}=0.75\), and \(n = 96\)

First, calculate the denominator \(\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.8\times(1 - 0.8)}{96}}=\sqrt{\frac{0.8\times0.2}{96}}=\sqrt{\frac{0.16}{96}}\approx\sqrt{0.001667}\approx0.0408\)

Then, \(z=\frac{0.75 - 0.8}{0.0408}=\frac{- 0.05}{0.0408}\approx - 1.22\)

Answer:

The test statistic \(z\approx - 1.22\)