# STATS – A biologist conducted an experiment to study the effect that various characteristics

August 30, 2017

Question
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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