Question

observation of GPA?​

Answer 1

Explanation:

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1. In a study of determinants of college GPA, a sample of 8 observations is examined. GPA is regressed on Achievement Scores (ACT), with an intercept. n (i) Let yi = GPAi, xi = ACTi, and n = 8. It is given that x = 25.875, y = 3.2125, ∑ (xi – i=1 n2ˆ x )(yi – y ) = 5.8125, and ∑ (xi – x ) = 56.875. Obtain the OLS estimate of the slope, β1 , i=1 rounded to four places after the decimal. Obtain an estimate of the intercept, βˆ0 . Are these estimators unbiased? Why and how? Does the intercept have a useful interpretation if ACT is not close to zero for the population of interest? What happens to GPA if ACT is 5 points higher?

(ii) The fitted values and residuals — rounded to four decimal places — are given along with the observation number i and GPA in the following table: 1 2.8 2.7143 .0857 2 3.4 3.0209 .3791 3 3.0 3.2253 –.2253 4 3.5 3.3275 .1725 5 3.6 3.5319 .0681 i GPA G P A uˆ 1 6 3.0 3.1231 –.1231 7 2.7 3.1231 –.4231 8 3.7 3.6341 .0659 You can verify that the residuals, as reported in the table, sum to −.0002, which is pretty close to zero. Is this what you would expect? Why? n

(iii) The sum of squared residuals, ∑uˆ2 , is about .4347 (rounded to four decimal places), n and the total sum of squares, ∑ (yi – y )2, is about 1.0288. Obtain the R-squared from this i i=1 i=1 regression. Justify the method/formula you employ to get the R-squared. What if there is no intercept in the regression? Is your computed R-squared value suggestive of a good “fit”? What may happen to R-squared if you add more regressors like “high school GPA” (and why?)?

(iv) If we add the hs-GPA to the above regression, for the full sample we obtain the following fitted regression: GPA=1.29+.453 hsGPA+.0094 ACT Describe the sense in which this regression allows us to quantify the ceteris paribus impact of hsGPA on college GPA. Describe, with careful arguments backed by known relations, what may happen if we now omit ACT from this regression. Note: The coefficient for hsGPA becomes .482, and sample correlation between hsGPA and ACT is .346.

Answer 2

Answer:

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