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 hypothesis. 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 \(p = 0.80\) (assuming no difference from the \(80\%\) claim), and the alternative hypothesis \(H_1\) is \(p<0.80\) (since we are testing for less than \(80\%\)).

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

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

Step3: Calculate the test - statistic \(z\)

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

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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 conclusion about the null hypothesis

Since the significance level \(\alpha = 0.01\) and \(P - value=0.236>0.01\), we fail to reject the null hypothesis \(H_0:p = 0.80\)

Step6: Make a final conclusion

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

Answer:

E. \(H_0:p = 0.80\), \(H_1:p<0.80\) (for the hypothesis part). Test statistic \(z\approx - 0.716\), \(P - value\approx0.236\). Fail to reject \(H_0\). There is not sufficient evidence to support the claim.