Question: A special program to help students make the transition from high school to the university provides an intense 5-week summer program prior to matriculation that focuses on the courses in mathematics and English that are required as well as study skills. The director of this program wants to demonstrate that those students who have participated in the program have a higher GPA at the end of their Freshman year (# of credit hours ? 18) than the general population of freshmen. He selects a random sample of 15 participants of the program and calculates their mean GPA to be 2.8 with a standard deviation of 0.5. He also selects a random sample of 15 freshmen from the general population of non-participants and calculates their mean GPA to be 2.2 with a standard deviation of 0.8

a. What assumptions must be met to use the t- distribution to analyze the differences in the means of these two populations of students?

b. Calculate the pooled sample estimator, S_p² .

c. Construct a 95% Confidence Interval for µ_{1 -} µ₂, the mean differences in GPAs for students enrolled in the special program and those in the general population.

d. Construct the Null and Alternative Hypothesis that the director would test to demonstrate that the mean GPA of students in the program is higher than the mean GPA of the general population of students.

e. What would be the Decision rule to test the Hypothesis at \(\alpha \) =.05.

f. Calculate the test statistic t^* and state the conclusion.