12-1 MULTIPLE LINEAR REGRESSION MODEL 423The slope estimate is about twice the magnitude of its standard error, and
are considerably larger than and This implies reasonable precision
of estimation, although the parameters 1 and 2 are much more precisely estimated than the
intercept (this is not unusual in multiple regression).EXERCISES FOR SECTION 12-1ˆ 1 and ˆ 2 se 1 ˆ 12 se 1 ˆ 22.se 1 ˆ 12 0.002798.12-1. A study was performed to investigate the shear
strength of soil (y) as it related to depth in feet (x 1 ) and mois-
ture content (x 2 ). Ten observations were collected, and the fol-
lowing summary quantities obtained: n10,and
(a) Set up the least squares normal equations for the model
Y 0 1 x 1 2 x 2 .
(b) Estimate the parameters in the model in part (a).
(c) What is the predicted strength when x 1 18 feet and
x 2 43%?
12-2. A regression model is to be developed for predicting
the ability of soil to absorb chemical contaminants. Ten obser-
vations have been taken on a soil absorption index (y) and two
regressors: x 1 amount of extractable iron ore and x 2
amount of bauxite. We wish to fit the model Y 0 1 x 1
2 x 2 . Some necessary quantities are:gy^2 i371,595.6.gxi 1 xi 2 12,352,gxi 1 yi43,550.8,gxi 2 yi104,736.8,gxi 2 553,gyi1,916,gxi^21 5,200.9,gxi^22 31,729,gxi 1 223,12-4. The data in Table 12-5 are the 1976 team performance
statistics for the teams in the National Football League
(Source: The Sporting News).
(a) Fit a multiple regression model relating the number
of games won to the teams’passing yardage (x 2 ), the percent
rushing plays (x 7 ), and the opponents’yards rushing (x 8 ).
(b) Estimate 2. 2
(c) What are the standard errors of the regression coefficients?
(d) Use the model to predict the number of games won when
x 2 2000 yards, x 7 60%, and x 8 1800.
12-5. Table 12-6 presents gasoline mileage performance for
25 automobiles (Source: Motor Trend,1975).
(a) Fit a multiple regression model relating gasoline mileage
to engine displacement (x 1 ) and number of carburetor
barrels (x 6 ).
(b) Estimate 2.
(c) Use the model developed in part (a) to predict mileage per-
formance for a car with displacement x 1 300 and x 6 2.(a) Estimate the regression coefficients in the model specified
above.
(b) What is the predicted value of the absorption index y
when x 1 200 and x 2 50?
12-3. A chemical engineer is investigating how the amount
of conversion of a product from a raw material (y) depends on
reaction temperature (x 1 ) and the reaction time (x 2 ). He has
developed the following regression models:
1.
2.
Both models have been built over the range 0.5 x 2 10.
(a) What is the predicted value of conversion when x 2 2?
Repeat this calculation for x 2 8. Draw a graph of the
predicted values for both conversion models. Comment
on the effect of the interaction term in model 2.
(b) Find the expected change in the mean conversion for a
unit change in temperature x 1 for model 1 when x 2 5.
Does this quantity depend on the specific value of reac-
tion time selected? Why?
(c) Find the expected change in the mean conversion for a
unit change in temperature x 1 for model 2 when x 2 5.
Repeat this calculation for x 2 2 and x 2 8. Does the
result depend on the value selected for x 2? Why?yˆ 95 1.5x 1 3 x 2 2 x 1 x 2yˆ 100 2 x 1 4 x 212-6. The electric power consumed each month by a chem-
ical plant is thought to be related to the average ambient
temperature (x 1 ), the number of days in the month (x 2 ), the
average product purity (x 3 ), and the tons of product produced
(x 4 ). The past year’s historical data are available and are pre-
sented in the following table:
yx 1 x 2 x 3 x 4
240 25 24 91 100
236 31 21 90 95
270 45 24 88 110
274 60 25 87 88
301 65 25 91 94
316 72 26 94 99
300 80 25 87 97
296 84 25 86 96
267 75 24 88 110
276 60 25 91 105
288 50 25 90 100
261 38 23 89 98(a) Fit a multiple linear regression model to these data.
(b) Estimate 2.1 ¿ 2 ^1 £1.17991 7.30982 E- 3 7.3006 E- 4
7.30982 E- 3 7.9799 E- 5 1.23713 E- 4
7.3006 E- 4 1.23713 E- 4 4.6576 E- 4§, ¿y£
220
36,768
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