Question

trials in an experiment with a polygraph include 96 results that include 22 cases of wrong results and 74 cases of correct results. use a 0.01 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
eq0.80 )
b. ( h_0: p = 0.80 )
( h_1: p>0.80 )
c. ( h_0: p = 0.20 )
( h_1: p
eq0.20 )
d. ( h_0: p = 0.20 )
( h_1: p>0.20 )
e. ( h_0: p = 0.80 )
( h_1: p<0.80 )
f. ( h_0: p = 0.20 )
( h_1: p<0.20 )

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 \(H_0\) is the statement of no difference, which is \(p = 0.80\) (assuming the proportion is 80% as a baseline for comparison). The alternative hypothesis \(H_1\) is the claim we are testing, \(p<0.80\)

Step2: Calculate the sample proportion \(\hat{p}\)

The sample size \(n = 96\), the number of correct results \(x = 74\). The sample proportion \(\hat{p}=\frac{x}{n}=\frac{74}{96}\approx0.7708\)

Step3: Calculate the test - statistic

The formula for the test - statistic \(z\) in a proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.80\), \(\hat{p}=0.7708\), \(n = 96\)

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Step4: Calculate the P - value

Since this is a left - tailed test (\(H_1:p < 0.80\)), the P - value is \(P(Z<z)\) where \(z\approx - 0.716\). Using a standard normal table or calculator, \(P(Z < - 0.716)\approx0.236\)

Step5: Make a decision about the null hypothesis

The significance level \(\alpha=0.01\). Since \(P - value=0.236>\alpha = 0.01\), we fail to reject the null hypothesis.

Answer:

The null hypothesis \(H_0:p = 0.80\) and the alternative hypothesis \(H_1:p < 0.80\) (corresponds to option E). The test statistic \(z\approx - 0.716\), the P - value\(\approx0.236\). Since \(P - value>\alpha\), we fail to reject \(H_0\). There is not sufficient evidence at the \(0.01\) significance level to support the claim that polygraph results are correct less than \(80\%\) of the time.