1. The director of a small college administered a newly designed entrance test to 20 students selected at random from the new freshman class. S/he wants to determine whether a student grade point average (GPA) at the end of the first year (Y) can be predicted from the entrance test score (X). The results are in file GPA and a brief description of the variables is in the file GPAd. Assume that the simple linear regression model (first-order model) is appropriate.

1. Obtain the least square estimates of b₀ and b₁ . State the estimated regression function.
2. Plot the regression function and the data. Does the estimated regression function appear to fit the data well?
3. Obtain a point estimate of the mean freshman GPA when the entrance test score is X=5.0.
4. What is the point estimate of the change in the mean response when the entrance test score increases by one point?

2. A person's muscle mass is expected to decrease with age. To explore this relationship in women, a study was performed with 16 women, selected such that there are four women in each 10-years age group, beginning with age 40 and ending with age 79. The results are in file MUSCLE and a brief description of the variables is in the file MUSCLEd. X is the age and Y is a measure of muscle mass. Assume that the simple linear regression model (first-order model) is appropriate.

1. Obtain the least square estimates of b₀ and b₁. State the estimated regression function.
2. Plot the regression function and the data. Does the estimated regression function appear to fit the data well? Does your plot support the anticipation that muscle mass decreases with age?
3. Obtain a point estimate of the mean muscle mass for women aged X=60 years.
4. Give the value of the residual for the eighth observation
5. Obtain a point estimate for s ²

3. A substance used in chemical research is shipped by air in cartons of 1,000 ampules. The data in file AIRFREIGHT involves 10 shipments. The collected data are on the number of times that the carton was transferred from one aircraft to another (X) and the number of ampules found broken upon arrival (Y). A brief description of the variables is in the file AIRFRd. Assume that the simple linear regression model (first-order model) is appropriate.

1. Obtain the estimated regression function. Plot the regression function and the data. Does the estimated regression function appear to fit the data well?
2. Obtain a point estimate of the expected number of broken ampules when X=1 transfers are made.
3. Estimate the increase in the the expected number of broken ampules when there are 2 transfers as compared to 1 transfer.
4. Verify that the fitted regression line goes through the point (X-bar, Y-bar)