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 area ≥ 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 and answer each question carefully, not just a report like
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SUMMARY OUTPUT
|
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|
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|
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Regression Statistics
|
|
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|
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Multiple R
|
0.981117
|
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R Square
|
0.96259
|
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Adjusted R Square
|
0.932662
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Standard Error
|
101.6917
|
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Observations
|
10
|
|
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ANOVA
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df
|
SS
|
MS
|
F
|
Significance F
|
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|
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Regression
|
4
|
1330434
|
332608.6
|
32.16344
|
0.000922
|
|
|
|
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Residual
|
5
|
51706.01
|
10341.2
|
|
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Total
|
9
|
1382141
|
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Coefficients
|
Standard Error
|
t Stat
|
P-value
|
Lower 95%
|
Upper 95%
|
Lower 95.0%
|
Upper 95.0%
|
|
Intercept
|
85.75037
|
125.1932
|
0.684944
|
0.523834
|
-236.069
|
407.5698
|
-236.069
|
407.5698
|
|
depth
|
-15.9333
|
5.142993
|
-3.09807
|
0.026912
|
-29.1538
|
-2.71286
|
-29.1538
|
-2.71286
|
|
cover
|
2.422796
|
1.6482
|
1.469965
|
0.201524
|
-1.81404
|
6.659628
|
-1.81404
|
6.659628
|
|
percent
|
1.827536
|
3.292229
|
0.555106
|
0.60274
|
-6.63541
|
10.29048
|
-6.63541
|
10.29048
|
|
area
|
3.073792
|
0.373856
|
8.221853
|
0.000433
|
2.112764
|
4.03482
|
2.112764
|
4.03482
|
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RESIDUAL OUTPUT
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Observation
|
Predicted biomass
|
Residuals
|
Standard Residuals
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1
|
64.08349
|
35.91651
|
0.473854
|
|
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2
|
368.0009
|
19.99915
|
0.263853
|
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3
|
844.2986
|
-89.2986
|
-1.17814
|
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4
|
1271.336
|
16.66447
|
0.219858
|
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5
|
194.7604
|
35.23964
|
0.464924
|
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6
|
126.8534
|
-126.853
|
-1.6736
|
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|
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7
|
478.7505
|
72.2495
|
0.953203
|
|
|
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|
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|
8
|
351.6054
|
-6.60541
|
-0.08715
|
|
|
|
|
|
|
9
|
74.19573
|
-74.1957
|
-0.97888
|
|
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|
10
|
231.1161
|
116.8839
|
1.542075
|
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