Question

The maintenance manager at a trucking company wants to build a regression model to forecast the time (in years) until the first engine overhaul based on four explanatory variables: (1) annual miles driven (in 1,000s of miles), (2) average load weight (in tons), (3) average driving speed (in mph), and (4) oil change interval (in 1,000s of miles). Based on driver logs and onboard computers, data have been obtained for a sample of 25 trucks. A portion of the data is shown in the accompanying table.

Time until First Engine Overhaul	Annual Miles Driven	Average Load Weight	Average Driving Speed	Oil Change Interval
7.6	43.0	24.0	45.0	20.0
0.8	98.9	28.0	45.0	28.0
⋮	⋮	⋮	⋮	⋮
5.9	60.7	25.0	53.0	23.0

b. Estimate the regression model. (Negative values should be indicated by a minus sign. Round your answers to 4 decimal places.)

TimeˆTime^  = ?+ ?Miles + ?Load + ?Speed + ?Oil

Time Until First Engine Overhaul	Annual Miles Driven	Average Load Weight	Average Driving Speed	Oil Change Interval
7.6	43	24	45	20
0.8	98.9	28	45	28
8.8	43.3	24	64	17
1.1	110.4	32	55	29
1.3	101.9	28	46	17
2.3	96.6	26	61	25
2.2	92.7	18	54	17
7.6	53.8	15	60	9
8.3	51.1	17	53	21
4.2	84.5	21	54	23
0.6	120	25	49	26
5.2	78	21	52	29
5.5	69.1	25	49	20
5.5	54.6	23	58	20
5.4	66.8	24	53	28
8.7	38.9	22	55	16
5.7	52.9	26	59	27
5.9	54.6	15	50	18
4.4	74.4	26	61	23
6.1	58.2	25	53	11
6.9	52.6	25	58	18
6.7	68.4	19	53	15
4.2	94.2	24	49	23
6.9	45.4	20	61	12
5.9	60.7	25	53	23

Answer 1

(b) The regression output of excel would be as below.

The regression output in R is as below.

-------------------------------------------------------------------

> # DATA INPUT
> library(readr)
> dat <- read_table2("dat.txt")

> # REGRESSION
> summary(lm(dat$Time ~ dat$Miles + dat$Weight + dat$Speed + dat$Oil))

Call:
lm(formula = dat$Time ~ dat$Miles + dat$Weight + dat$Speed +
    dat$Oil)

Residuals:
     Min       1Q   Median       3Q      Max
-1.26984 -0.41224 0.01954 0.42892 1.43511

Coefficients:
             Estimate Std. Error t value Pr(>|t|)  
(Intercept) 12.923142   2.306600   5.603 1.75e-05 ***
dat$Miles   -0.094872   0.009320 -10.179 2.35e-09 ***
dat$Weight -0.071587   0.050772 -1.410    0.174  
dat$Speed    0.009589   0.035247   0.272    0.788  
dat$Oil      0.001170   0.038466   0.030    0.976  
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8449 on 20 degrees of freedom
Multiple R-squared: 0.9042,   Adjusted R-squared: 0.885
F-statistic: 47.19 on 4 and 20 DF, p-value: 6.535e-10

-------------------------------------------------------------------

The estimated model is hence

,

and rounding to 4 decimals, we have

.

(d) For the given values, we have

(miles in thousands)

or

units (years). This means that for the given values, the expected time until first engine overhauls is 5.94 years.