A biologist conducted an experiment to
study the effect that various characteristics of a stream have on the amount of
fish biomass that the stream supports. The regressor variables are as follows:
X1: average depth of stream
X2: area of instream cover
(i.e., undercut banks, logs, boulders, etc.)
X3: percent canopy cover
X4: amount of surface area³ 25 cm in
depth.
The response variable is y, the fish
biomass.
Data file: biomass.
biomass
depth
cover
percent
area
100
14.3
15
12.2
48
388
19.1
29.4
26
152.2
755
54.6
58
24.2
469.7
1288
28.8
42.6
26.1
485.9
230
16.1
15.9
31.6
87.6
0
10
56.4
23.3
6.9
551
28.5
95.1
13
192.9
345
13.8
60.6
7.5
105.8
0
10.7
35.2
40.3
0
348
25.9
52
40.3
116.6
(a) Fit the full model with all 4 regressor
variables to these data. Find S2 and R2
.
(b) Perform the overall F-test for this
model, explicitly stating the null and alternative hypotheses corresponding to
this F-test, and state your conclusions.
(c) Compute t-statistics used to test
whether each of the regression coefficients in this model individually is equal
to zero, and state your conclusions.
(d) Now fit the model y =b0 +b2X2 +b2X4 +e. If we define this model as the “reduced model” and the model in
part (a) as the “full model”, what are the null and alternative hypotheses that
will be tested by the F-test corresponding to the ANOVA whose residual SS is
that from the full model and whose total SS is the residual SS from the reduced
model? Perform this test and state your conclusions.
Please include R or SAS
code, if you use Minitab, include the procedure.
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