Question

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. (i) 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. 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 ?

Answer 1

To estimate the relationship between GPA and ACT scores using OLS, a regression analysis would be performed. The intercept may not have a practical interpretation since ACT scores are not zero, and to predict GPA increase from a five-point ACT score increase, multiply the slope estimate by five. For an ACT score of 20, use the estimated regression equation to find the predicted GPA.

Step-by-step explanation:

To estimate the relationship between GPA and ACT scores using OLS (Ordinary Least Squares), we would perform a regression analysis where GPA is the dependent variable and ACT is the independent variable. The regression equation would take the form GPA = β⁰ + β¹ ACT, where β⁰ is the intercept and β¹ is the slope of the line. The direction of the relationship can be determined by the sign of the slope estimate (β¹). Typically, we would expect a positive relationship, meaning as ACT scores increase, so does the GPA. The intercept (β⁰) represents the expected GPA when the ACT score is zero. While this might not be a practical scenario (as ACT scores are never zero), it is a necessary part of the regression equation.

If the ACT score is increased by five points, the predicted increase in GPA would be five times the slope estimate (β¹). To compute the fitted values and residuals for each observation, we would use the estimated regression equation, with residuals being the differences between the actual GPA values and the predicted values from the regression. The sum of the residuals should approximately equal zero if the model is correctly specified.

For predicting the GPA when the ACT score is 20, we would plug in 20 for the ACT score in our regression equation and calculate the corresponding GPA.