3 The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 5 3.6 29 6 3.0 25 7 2.7 25 8 3.7 30 (i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation GPA = ??0 + ??1ACT. Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? (ii) Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum to zero. (iii) What is the predicted value of GPA when ACT = 20? (iv) How much of the variation in GPA for these eight students is explained by ACT? Explain.
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The slope (β1) is the change in the dependent variable (GPA) for a one-unit change in the independent variable (ACT). The intercept (β0) is the value of the dependent variable when the independent variable is zero. To calculate these, we need to use the formulas Show more…
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The following table contains the ACT scores and the GPA (grade point average) for eight college students. GPA is based on a four-point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 5 3.6 29 6 3.0 25 7 2.7 25 8 3.7 30 i. Estimate the equation GPA_i = ̣̂̂β_0 + ̂β_1ACT_i + ε_i using OLS. (5M) ii. Calculate their variance and standard error. (5M) iii. Comment on the direction of the relationship. (2M) iv. Does the intercept have a useful interpretation here? Explain. (3M) v. How much higher is the GPA predicted to be if the ACT score is increased by five points? (3M) vi. Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum is zero. (6M) vii. What is the predicted value of GPA when ACT=20? (2M) viii. How much the variation in GPA for each eight students is explained by ACT? (4M)
Adi S.
4) The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 5 3.6 29 6 3.0 25 7 2.7 25 8 3.7 30 Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation (GPA) ̂=β ̂_0+β ̂_1 ACT+u Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? Compute the fitted values and residuals for each observation and verify that the residuals (approximately) sum to zero. What is the predicted value of GPA when ACT= 20? How much of the variation in GPA for these eight students is explained by ACT? Explain.
Jon S.
Use the data in TWOYEAR for this exercise. (i) The variable stotal is a standardized test variable, which can act as a proxy variable for unobserved ability. Find the sample mean and standard deviation of stotal. (ii) Run simple regressions of jc and univ on stotal. Are both college education variables statistically related to stotal? Explain. (iii) Add stotal to equation (4.17) and test the hypothesis that the returns to two- and four-year colleges are the same against the alternative that the return to four-year colleges is greater. How do your findings compare with those from Section 4-4? (iv) Add stotall to the equation estimated in part (iii). Does a quadratic in the test score variable seem necessary? (v) Add the interaction terms stotal jc and stotal univ to the equation from part (iii). Are these terms jointly significant? (vi) What would be your final model that controls for ability through the use of stotal? Justify your answer.
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