The maintenance manager at a trucking company wants to build a regression model to forecast the time until the first engine overhaul based on four explanatory variables: (1) annual miles driven, (2) average load weight, (3) average driving speed, and (4) oil change interval. Based on driver logs and onboard computers, data have been obtained for a sample of 25 trucks.

Time Until First Engine Overhaul (Yrs) Annual Miles Driven (000) Average Load Weight (tons) Average Driving Speed (mph) Oil Change Interval (000 miles)
7.9 42.8 19 46 15
0.9 98.5 25 46 29
8.5 43.4 21 64 14
1.3 110.7 27 60 26
1.4 102.3 28 51 17
2.1 97.1 24 63 20
2.5 92.8 23 55 15
7.4 53.9 20 65 13
8.2 51.4 22 52 17
4.1 84.9 25 56 28
0.5 120.4 29 52 23
5.1 77.5 25 48 27
5.2 68.6 21 48 25
5.3 54.9 24 58 23
5.7 66.7 20 58 26
8.5 39.4 20 50 16
5.8 52.7 21 56 25
5.9 54.2 19 48 17
4.4 74.8 22 65 25
6.3 58.7 20 54 16
6.7 52.3 22 53 19
7.0 68.6 18 51 19
3.9 94.6 23 54 23
7.2 45.7 17 58 15
6.1 61.2 24 58 19

a. Estimate the regression model to predict the time before the first engine overhaul for a truck driven 60,000 miles per year with an average load of 22 tons, an average driving speed of 57 mph, and 18,000 miles between oil changes. (Note that both annual miles driven and oil change interval are measured in 1,000s.)

b. Use the above prediction to calculate and interpret the 90% confidence interval for the mean time before the first engine overhaul.

c. Calculate and interpret the corresponding 90% prediction interval for the time before the first.