Trials in an experiment with a polygraph include 99, results that include 23, ca

Question

Trials in an experiment with a polygraph include 99, results that include 23, cases of wrong results and 76 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.

1)

A. H0: p =0.20 H1:p 0.20

B. H0: p =0.80 H1: < 0.80

C. H0: p =0.20 H1: p > 0.20

D. H0: p =0.80 H1: p > 0.80

E. H0: p=0.80 H1: p 0.80

F. H0: p = 0.20 H1: p < 0.20

2) The test statistic is z = ???

3) The p value is???

4) Identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim. [ (A) Reject or (B) Fail to reject] Ho. There [(A) is not , or (B) is ] sufficient evidence to support the claim that the polygraph results are correct less than 80% of the time.

Explanation / Answer

Solution:-

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P = 0.80
Alternative hypothesis: P < 0.80

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected only if the sample proportion is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation () and compute the z-score test statistic (z).

= sqrt[ P * ( 1 - P ) / n ] = sqrt [(0.8 * 0.2) / 100] = sqrt(0.0016) = 0.04

Population proportion for correct answers, p = 76/99 = 0.77
z = (p - P) / = (0.77 - 0.80)/0.04 = -0.75

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -0.75. We use the Normal Distribution Calculator to find P(z < -0.75) = 0.226627.

Thus, the P-value = 0.226627.

Interpret results. Since the P-value (0.226627) is greater than the significance level (0.01), we can accept the null hypothesis.

Conclusion. Fail to Reject the null hypothesis. There is not sufficient evidence to support the claim that the polygraph results are correct less than 80% of the time.

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