TABLE 12-5
A corporation randomly selects 150 salespeople and finds that 66% who have never taken a self-improvement course would like such a course. The firm did a similar study 10 years ago in which 60% of a random sample of 160 salespeople wanted a self-improvement course. The groups are assumed to be independent random samples. Let π 1 and π 2 represent the true proportion of workers who would like to attend a self-improvement course in the recent study and the past study, respectively.
-Referring to Table 12-5, the company tests to determine at the 0.05 level whether the population proportion has changed from the previous study. Which of the following is most correct?
A) Reject the null hypothesis and conclude that the proportion of employees who are interested in a self-improvement course has changed over the intervening 10 years.
B) Do not reject the null hypothesis and conclude that the proportion of employees who are interested in a self-improvement course has not changed over the intervening 10 years.
C) Reject the null hypothesis and conclude that the proportion of employees who are interested in a self-improvement course has increased over the intervening 10 years.
D) Do not reject the null hypothesis and conclude that the proportion of employees who are interested in a self-improvement course has increased over the intervening 10 years.

Question

Quizplus · Accepted Answer

We want to conduct a hypothesis test to determine if the population proportion has changed from the previous study. Let p1 and p2 represent the proportions in the recent study and the past study, respectively. Our null hypothesis is that there is no difference between the proportions, so H0: p1 = p2. Our alternative hypothesis is that there is a difference between the proportions, so Ha: p1 ≠ p2.

We are given that n1 = 150 and n2 = 160, and that the samples are independent random samples. We can assume that the conditions for inference are met since the sample sizes are large enough and the observations are independent.

We can use a two-sample z-test to test this hypothesis. The test statistic is:

z = (p1 - p2) / sqrt( (p̂1(1 - p̂1) / n1) + (p̂2(1 - p̂2) / n2) )

where p̂1 and p̂2 are the sample proportions.

Using the given information, we have:

p̂1 = 0.66
p̂2 = 0.60
n1 = 150
n2 = 160

Plugging these values into the formula, we get:

z = (0.66 - 0.60) / sqrt( (0.66(1 - 0.66) / 150) + (0.60(1 - 0.60) / 160) ) = 1.15

Using a standard normal distribution table or calculator, we find the p-value to be 0.25 (two-tailed test). Since the p-value is greater than the significance level of 0.05, we do not reject the null hypothesis. Therefore, we conclude that there is not enough evidence to suggest that the proportion of employees who are interested in a self-improvement course has changed over the intervening 10 years. Thus, the correct answer is B.

TABLE 12-5 A Corporation Randomly Selects 150 Salespeople and Finds That 66