SCENARIO 12-1
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 $\pi$ 1 and $\pi$ 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 Scenario 12-1,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

SCENARIO 12-1
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  $\pi$ 1 and  $\pi$ 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 Scenario 12-1,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.

Quizplus · Accepted Answer

The null hypothesis is that there is no change in the proportion of employees who are interested in a self-improvement course over the 10 years. The alternative hypothesis is that there is a change.
H0: F1=F10
HA: F1≠F10
We can use a two-sample z-test for proportions to test this hypothesis. 
z = (p̂1 - p̂2) / sqrt(p(1-p)(1/n1 + 1/n2))
where p̂1 is the sample proportion from the recent study, p̂2 is the sample proportion from the past study, p is the pooled proportion, n1 and n2 are the sample sizes from the recent and past studies respectively.

p̂1 = 0.66
p̂2 = 0.6
n1 = 150
n2 = 160
p = (x1 + x2) / (n1 + n2) = (0.66*150 + 0.6*160) / (150 + 160) = 0.629

z = (0.66 - 0.6) / sqrt(0.629(1-0.629)(1/150 + 1/160)) = 1.34

At α = 0.05 level of significance, the critical value for a two-tailed test is ±1.96. Since the calculated z-score of 1.34 falls within the acceptance region (-1.96 to +1.96), we cannot reject the null hypothesis and conclude that there is not enough evidence to suggest a change in the proportion of employees who are interested in a self-improvement course over the 10 years. Therefore, the most correct choice is B, which states that we 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.

SCENARIO 12-1 a Corporation Randomly Selects 150 Salespeople and Finds π\piπ

SCENARIO 12-1
a Corporation Randomly Selects 150 Salespeople and Finds $\pi$