STAT 112 --- Quiz 5 --- Solutions
3/10/04
An article published in Geography used multiple regression to predict annual rainfall levels in
California. Data on the annual average precipitation (y), altitude (x1), latitude (x2), and distance
from the Pacific Coast (x3) for 30 meteorological stations scattered throughout California were
collected.
The model y   0  1 x1   2 x2   3 x3   was fit to the data.
The following is the SAS output for the regression of y on the three predictor variables:
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Total
3
26
29
4809.35596
3202.29762
8011.65359
Root MSE
Dependent Mean
Coeff Var
F Value
Pr > F
1603.11865
123.16529
13.02
<.0001
R-Square
Adj R-Sq
0.6003
0.5542
11.09799
19.80733
56.02968
Parameter Estimates
Variable
Intercept
altitude
latitude
distance
DF
Parameter
Estimate
Standard
Error
t Value
1
1
1
1
-102.35743
0.00409
3.45108
-0.14286
29.20548
0.00122
0.79486
0.03634
-3.50
3.36
4.34
-3.93
Pr > |t|
0.0017
0.0024
0.0002
0.0006
a. (6 pts) State and test the null for the overall utility of the model. Use α=0.05.
H0: β1= β2 = β3=0
versus H1: at least one is non-zero.
To test this hypothesis, we use the F-test statistic defined by
SS Re g
F 
k
SSE
n  (k  1)

MS Re g
MSE
 13.02
The value of the F-test statistic is given in the ANOVA table. The p-value of the test is
smaller than .0001, which is smaller than .05. Hence we reject the null in favor of the
alternative and we find the model to be useful.
b. (6 pts) Does distance from the coast affect the precipitation? α=0.01.
This is equivalent to testing H0: β3=0 versus H1: β3≠ 0. This is a t-test. The t-test
statistic has value -3.93 and the p-value of the test is 0.0006 < .01. Hence we reject the
null in favor of the alternative to conclude that distance from the coast has a statistically
significant effect on precipitation.
c. (4 pts) What is the value of the estimate of β2, the coefficient of latitude?
What is its meaning?
The estimate of β2 is 3.45108 and represents the increase in precipitation if latitude
is increased by 1 unit while both altitude and distance are held fixed.
d. (4 pts) 95% prediction intervals for y are attached. Locate and interpret the interval for
the Giant Forest meteorological station (station #9).
The 95% prediction interval for the precipitation at the Giant Forest meteorological
station is (3.7054, 54.8014). Hence we expect that with probability 95% this interval
covers the true exact annual average precipitation at the Giant Forest
meteorological station.