Question

A university administrator claims that, at their university, juniors are less likely than freshmen to take summer classes. They select a random sample of 92 juniors and 85 freshmen at their university, and find that 31 juniors and 37 freshmen take summer classes. Test the university administrator’s claim at a 10% significance level.

a) Define Population 1 and Population 2.

b) Define the parameter of interest.

c) State the null and alternative hypotheses.

d) Assume the test statistic has a Z-distribution. Calculate the test statistic.

e) State your decision. Can you support the university administrator’s claim at a 10% significance

level?

Answer 1

To test the university administrator's claim, we define the populations, the parameter of interest, and the null and alternative hypotheses. Then, we calculate the test statistic and compare it to the critical value to make a decision. At a 10% significance level, we determine whether we can support the administrator's claim or not.

Step-by-step explanation:

a) Population 1: Juniors at the university
Population 2: Freshmen at the university

b) The parameter of interest is the proportion of students who take summer classes.

c) Null hypothesis (H0): The proportion of juniors taking summer classes is the same as the proportion of freshmen taking summer classes.
Alternative hypothesis (Ha): The proportion of juniors taking summer classes is less than the proportion of freshmen taking summer classes.

d) To calculate the test statistic, use the formula:
Z = (pˆ1 - pˆ2) / sqrt((pˆ(1-pˆ))(1/n1 + 1/n2))
Where pˆ1 is the proportion of juniors taking summer classes, pˆ2 is the proportion of freshmen taking summer classes, n1 is the sample size of juniors, and n2 is the sample size of freshmen.

e) The decision is based on the critical value of the test statistic. If the test statistic is less than the critical value, we reject the null hypothesis. If the test statistic is greater than or equal to the critical value, we fail to reject the null hypothesis.

f) Based on the test statistic calculated in part d, compare it to the critical value for a 10% significance level. If the test statistic is less than the critical value, we can support the university administrator's claim that juniors are less likely than freshmen to take summer classes. If the test statistic is greater than or equal to the critical value, we cannot support the claim.