Using the datain GPA2RAW the following equation was estimate

Using the datain GPA2RAW, the following equation was estimated at= 1,028.10 + 19.30 hsize-219 hsi ize 45.09 female (6.29) (3.83) (.53) (4.29) -169.81 black + 62.31 female black (12.7 (18.15) 4, l 37, R-.0858. n The variable sat is the combined SAT score, hsize is size of the student\'s high school gra cuating class, in hundreds, fema /e is a gender dummy variable, and b/ack is arace dummy variable equal to one f 1) Is there strong evidence that hsize should be included in the model? 2) True or False: from this equation, the high school size have diminishing marginal effect on SAT or blacks and zero otherwise score 3) Whenthe hsize has negative marginal effect on SA T? 4) Holding hsie fixed, whatis the estimated difference in SAT score between nonblack females and nonblack males? Is this estimated difference statistically significant? 5) What is the estimated difference in SAT score between nonblack males and black males? 6) Whatis the estimated difference in SAT score between black females and nonblack females?

Solution

1)

Let us calculate the t-statistic for hsize².

t = (-2.19 – 0)/0.53 = - 4.13

This will easily be greater than 2.5 which is the critical value of t test at 1% level of significance.

2) TRUE

Let us find first derivative of the regression equation:

dsat = 19.30 – 2.19*2* hsize*dhsize
dsat = 19.30 – 4.38* hsize* dhsize

As we can see, when hsize increases by 1 unit, the change in predicted sat score decreases by 4.38. It means that predicted sat score exhibits diminishing marginal effect with respect to hsize.

3)

For this part, we must solve for hsize where dsat =0.
19.30 – 4.38* hsize* dhsize = 0

Take dhsize =1, then –
4.38*hsize = 19.30
hsize = 4.41

That is marginal effect of hsize will be 0 when hsize = 4.41.

In other words, it will be negative whenever hsize exceeds 4.41.

4)

We hold hsize = 0 here.
Black dummy = 0 since we will deal with non-blacks.
gender dummy = 1 for females and gender dummy = 0 for males.

Then predicted sat for females = 1028.1 – 45.09*1 – 169.81*0 + 62.31*1*0 = 983.01

Then predicted sat for males = 1028.1 – 45.09*0 – 169.81*0 + 62.31*0*0 = 1028.1

It can be verified that coefficient of all dummies are statistically significant and hence it will be the case that the estimated difference is statistically significant

PS: According to chegg guidelines in event of multiple parts, only first four are attempted. However, you may repost it asking for solution of specific questions/parts.