Question

The following equation has been estimated:

=1028.10+19.3 hsize-1.90 hsize_squared-53.29 female-156.26 black+37.87 female*black

(6.29) (3.83) (0.53) (4.29) (14.66) (18.15)

The variable sat is the combined SAT score, hsize is the size of the students' high school graduating class in hundreds. The variable female is a gender dummy variable equal to 1 for females and 0 otherwise. Another variable black is a race dummy variable equal to 1 for black and 0 otherwise. Robust standard error in the parentheses.

(1)  Holding hsize fixed, what is the estimated difference in SAT score between nonblack males and black males?

(2) Is this estimated difference calculated in (1) statistically significant from 0? Calculate the t statistics.

(3) How statistically significant is this estimated difference?

(4) Holding hsize fixed, what is the estimated difference in SAT score between black females and nonblack male?

Could you please type it?

Answer 1

1) Since hsize is fixed, assume hsize =0

SAT score of nonblack male is when gender dummy variable and race dummy variable are both zero.

SAT Score(nonblack male) = 1028.10 - 53.29*0- 156.26*0 + 37.87*0

=1028.10

SAT score for black male when gender dummy=0 and race dummy =1

SAT Score (black male) = 1028.10 - 53.29*0 - 156.26*1 + 37.87*0

= 1028.10 - 156.26 = 871.84

So ,estimated difference in SAT score = SAT Score(black male) - SAT Score(nonblack male)= 871.84 - 1028.10 = (-156.26)

2) The estimated difference is same as the coefficient of race dummy variable (-156.26).

So t statistic can be calculated using the coefficient and standard error of the race dummy variable.

t = coefficient÷standard error of the coefficient

= (-156.26) ÷ 14.66 = (-10.65)

Such a high t value in absolute term is generally statistically significant.

3) In case of a standard normal distribution z< -1.96 and z> 1.96 mark the cutoff for statistical significance if α is set at 5%. So if sample size is big enough,then a t value is significant if the absolute t value is higher or equal to 1.96 (|t|≥1.96).

Here, as |t|=10.65 > 1.96 (assuming sample is big) ,hence the estimated difference is statistically significant.

4) SAT Score of black female is when both gender dummy and race dummy are equal to 1.

SAT Score(black female) = 1028.10 - 53.29 - 156.26 + 37.87 = 856.42

SAT score of nonblack male is when both dummy variables are 0.

SAT Score(nonblack male) = 1028.10

Estimated difference in SAT score = 1028.10 - 856.42 =171.68

So, the mean SAT score of nonblack males is higher than score of black females by 171.68