Question

causally! What might be going on C12 Use the data in ECONMATH to answer the following questions. 0 Estimate a model explaining colgpa to hsgpa, actmth, and acteng. Report the results in the usua form. Are all explanatory variables statistically significant? (ii) Consider an increase in hsgpa of one standard deviation, about .343. By how much does colgpo increase, holding actmth and acteng fixed. About how many standard deviations would the actmth have to increase to change colgpa by the same amount as a one standard deviation in hsgpa? Comment. (ii) Test the null hypothesis that actmth and acteng have the same effect (in the population) against a two-sided alternative. Report the p-value and describe your conclusions. (iv) Suppose the college admissions officer wants you to use the data on the variables in part (i) to create an equation that explains at least 50 percent of the variation in colgpa. What would you tell the officer? reg colgpa hsgpa actmth acteng 814 Number of obs Source df MS F(3, 810) 92.76 Model 60.4406735 3 20.1468912 Prob > F 0.0000 Residual 810.217187195 R-squared 175.921628 0.2557 Adj R-squared0.2530 236.362301 Total 813 .290728538 Root MSE 46603 Coef. [95% Interval] colgpa Std. Err. Conf. 053046 hsgpa .5548717 6590007 12.42 0.000 .7631245 0130198 actmth 0051539 2.53 0.012 . 0029033 . 0231364 012219 0050443 2.42 0.016 .0023176 .0221203 acteng 0282369. . 3009652 .3574391 .1677124 0.17 0.866 cons sum Variable Obs Std. Dev. Min Mean Max 856 .9369125 18 29 19.40654 age work 856 8.625584 37.5 9.2285 50 study 856 7.804398 13.91799 856 econhs 3703271 . 4831746 1 856 875 colgpa . 5368669 2.814686 4 4.26 hsgpa 856 3.34243 .3434283 2.355 12 34 814 22.59459 3.788735 acteng 814 23.211.3 12 actmth 3.773354 36 814 13 act 23.12162 3.349094 856 10 mathscr 7.875 1.731587 male 856 .5002923 1 856 calculus 6764019 .4681222 1 856 296729 attexd .4570831 1 attgood 856 5864486 .4927578 fathcoll 856 5245327 .4996897 1 mothcoll 856 6285047 . 483487 1 856 19.53 72.59981 13.40068 98.44 score

Answer 1

(i)
The regression model is,
colgpa = 0.0282369 + 0.6590007 hspgpa + 0.0130198 actmth + 0.012219 acteng

P-value of all the coefficients are less than the significance level 0f 0.05. Thus all explanatory variables are significant.

(ii)
The slope of hsgpa is 0.6590007
Keeping actmth and acteng fixed, increase in colgpa with increase of 0.343 in hspgpa = 0.343 * 0.6590007 = 0.226

The slope of actmth is 0.0130198.
Let X be the increase in actmth which led to increase of 0.226 in colgpa.
Then 0.0130198 * X = 0.226
=> X = 0.226 / 0.0130198 = 17.35818
Number of std deviations of X = 17.35818 / 3.77 = 4.6
We need to increase actmth by 4.6 std deviations to get the same increase in colgpa as with increase in hspgpa by one std deviations.

(iii)

Ho: Bactmth Bacteng

Ha Pactmth

Assuming covariance between coefficients of actmth and acteng is 0,

SE(
actmth Bacteng
) = $\sqrt{ SE(\beta_{actmth})^2 + SE(\beta_{acteng})^2 } = \sqrt{0.00515^2 + 0.0504^2}$

= 0.05066

Test statistic, t = (0.0130198 - 0.012219) / 0.05066

= 0.0158

Df = 810

P-value = 2 * P(t > 0.0158, df = 810) = 0.9874

Since, p-value is greater than the significance level of 0.05, we reject H0 and conclude that there is no evidence to reject the claim that actmth and acteng have the same effect on colgpa.

(iv)

To increase the explained variance (R²), you need to run the regression on more number of observations, so that the regression model more accurately fits the data and minimize the SSE.