Universities are not allowed to give athletic scholarships. Suppose there is a s

Question

Universities are not allowed to give athletic scholarships. Suppose there is a suspicion that at Willard Leclerc University,1 athletes are given more attractive financial aid packages than non-athletes. Data for 127 students at WLU for the following variables was secretly collected by the crack team of CBC’s The Fifth Estate.

• FAi = financial aid of students i
• GPAi = high school grade point average of student i
• STVi = Standardized Test verbal score of student i
• STMi = Standardized Test math score of student i
• Ai = 1 if the student is an athlete and 0 if not (a dummy variable)

There are two potential sources of favoritism. First, financial aid packages could be more generous for athletes (holding high school GPA and ST scores constant). Second, financial aid decisions could be made separately for athletes and non-athletes.

To test the first possibility, the CBC’s crack team of investigators estimated the following regression by OLS. (standard errors are reported in parenthesis below the coe cient).

FAi = 4.36 + 2.3 GPAi + 0.04STVi + 0.06 STMi + 1.237Ai

(3.02) (1.15) (0.02) (0.025) (0.5) R^2 = 0.42

RSS = 0.062

(a) How would you interpret the estimate of the coefficient on Ai?

(b) Test individually the hypothesis that each of the slope coefficients are zero at the 5% significance level. Is there evidence that athletes are getting favors in terms of financial aid packages?

(c) Suppose a CBC staffer with an undergraduate degree from Waterloo University placed the variable “Total STA score” in the original regression in addition to the variables already there. Is that a good estimating idea? How would that affect the OLS regression?

Explanation / Answer

Solution:

A) Based on the equation, it is evident that the financial aid for the athletic students is clearly more by an amout of 1.237.

B) We have to test that slope coefficients is 0. Lets assume that null hyothesis is that slope is not 0. If we can find enough evidence, we will be unable to reject the null hypothesis and then can justify that the slope is not 0.

Let's calculate the Standard Error(SE) and degrees of freedom(DF) for each slope:

GPAi: SE = 1.15 ; DF = Sample Size(n) - 2 = 127 - 2 = 125

We are subtracting 2 as we have to test the slopes individually. So, 1 dependent and 1 independent.

Now let's calculate the t statistics for the slope:

t = Slope(GPAi) / SE(GPAi)
= 2.3/1.15 = 2

The pvalue corresponding to the t = 2 and DF = 125 is 0.047 which is less than the significance level 0.05, so we reject the null hypothesis. I.e., the slope of GPAi is 0.

Similary for STVi:

t = Slope(STVi) / SE(STVi)
= 0.04/0.02 = 2 with 125 degrees of freedom

p value is 0.047, rejecting the null hypothesis. Slope is 0.

STMi: t = Slope(STMi) / SE(STMi)
= 0.06/0.025 = 2.4 with 125 degree of freedom

p value is 0.01787, rejecting the null hypothesis. Slope is 0.

Ai: t = Slope(Ai) / SE(Ai)
= 1.237/0.5 = 2.474 with 125 degree of freedom

p  value is 0.0147, rejecting the null hypothesis. Slope is 0.

C) No, it wouldn't be a good idea. As it would be higly correlated with the STV and STM score.It will inflate standard errors of the estimate which will result in large confidence intervals and may lead to type 2 error. Basically, we will end up accepting too many null hypothesis.

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