Question

Trials in an experiment with a polygraph include 98 results that include 23 cases of wrong results and 75 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.

Answer 1

Solution:

Here, we have to use one sample z test for the population proportion.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H0: The polygraph results are correct 80​% of the time.

Alternative hypothesis: Ha: The polygraph results are correct less than 80​% of the time.

H0: p = 0.8 versus Ha: p < 0.8

This is a lower tailed test.

We are given

Level of significance = α = 0.05

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 75

n = sample size = 98

p̂ = x/n = 75/98 = 0.765306122

p = 0.8

q = 1 - p = 0.2

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.765306122 – 0.8)/sqrt(0.8*0.2/98)

Z = -0.8586

Test statistic = -0.8586

P-value = 0.1953

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the polygraph results are correct less than 80​% of the time.