Question

Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Lie Detectors Trials in an experiment with a polygraph yield 98 results that include 24 cases of wrong results and 74 cases of correct results (based on data from experiments conducted by researchers Charles R. Honts of Boise State University and Gordon H. Barland of the Department of Defense Polygraph Institute). Use a 0.05 significance level to test the claim that such polygraph results are correct less than 80% of the time. Based on the results, should polygraph test results be prohibited as evidence in trials?

Short answer

Null hypothesis: The proportion of correct results is equal to 80%.

Alternative hypothesis: The proportion of correct results is less than 80%.

Test statistic: -1.111

Critical value: -1.645

P-value: 0.1332

The null hypothesis is failed to reject.

There is not enough evidence to support the claim that the proportion of correct results is less than 80%.

The sample proportion of correct polygraph results equal to 75.5% is not very high, and thus, polygraph test results should be prohibited as evidence in trials as they are not reliable enough.

Step-by-step solution

Given information

Out of 98 polygraph results, 74 are correct results, and 24 are wrong results.

Hypotheses

The null hypothesis is written as follows.

The proportion of correct results is equal to 80%.

$H_0:p=0.80$

The alternative hypothesis is written as follows.

The proportion of correct results is less than 80%.

$H_1:p<0.80$

The test is left-tailed.

Sample size, sample proportion, and population proportion

The sample size is n=98.

The sample proportion of correct results is computed below.

$$\begin{gathered} \hat{p}=\frac{\text{Number}\text{of}\text{correct}\text{results}}{\text{Sample}\text{Size}} \\ =\frac{74}{98} \\ =0.755 \end{gathered}$$

The population proportion of correct results is equal to 0.80.

Test statistic

The value of the test statistic is computed below.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.755-0.80}{\sqrt{\frac{0.80\left(1-0.80\right)}{98}}} \\ =-1.111 \end{gathered}$$

Thus, z=-1.111.

Critical value and p-value

Referring to the standard normal table, the critical value of z at $\alpha =0.05$ for a left-tailed test is equal to -1.645.

Referring to the standard normal table, the p-value for the test statistic value of -1.111 is equal to 0.1332.

As the p-value is greater than 0.05, the null hypothesis is failed to reject.

Conclusion of the test

There is not enough evidence to support the claim that the proportion of correct results is less than 80%.

The sample proportion of correct polygraph results equal to 75.5% is not very high. Thus, polygraph results do not appear to have a high accuracy rate that is required to produce them in court.

Thus, their use in the court as evidence should be ceased.