Reference no: EM13917063

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

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.981117
R Square	0.96259
Adjusted R Square	0.932662
Standard Error	101.6917
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	4	1330434	332608.6	32.16344	0.000922
Residual	5	51706.01	10341.2
Total	9	1382141
	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
RESIDUAL OUTPUT
Observation	Predicted biomass	Residuals	Standard Residuals
1	64.08349	35.91651	0.473854
2	368.0009	19.99915	0.263853
3	844.2986	-89.2986	-1.17814
4	1271.336	16.66447	0.219858
5	194.7604	35.23964	0.464924
6	126.8534	-126.853	-1.6736
7	478.7505	72.2495	0.953203
8	351.6054	-6.60541	-0.08715
9	74.19573	-74.1957	-0.97888
10	231.1161	116.8839	1.542075