Question

The length of a species of fish is to be represented as a function of the age (measured in days) and water temperature (degrees Celsius). The fish are kept in tanks at 25, 27, 29 and 31 degrees Celsius. After birth, a test specimen is chosen at random every 14 days and its length measured.

	Age	Temp	Length
1	14	25	620
2	28	25	1,315
3	41	25	2,120
4	55	25	2,600
5	69	25	3,110
6	83	25	3,535
7	97	25	3,935
8	111	25	4,465
9	125	25	4,530
10	139	25	4,570
11	153	25	4,600
12	14	27	625
13	28	27	1,215
14	41	27	2,110
15	55	27	2,805
16	69	27	3,255
17	83	27	4,015
18	97	27	4,315
19	111	27	4,495
20	125	27	4,535
21	139	27	4,600
22	153	27	4,600
23	14	29	590
24	28	29	1,305
25	41	29	2,140
26	55	29	2,890
27	69	29	3,920
28	83	29	3,920
29	97	29	4,515
30	111	29	4,520
31	125	29	4,525
32	139	29	4,565
33	153	29	4,566
34	14	31	590
35	28	31	1,205
36	41	31	1,915
37	55	31	2,140
38	69	31	2,710
39	83	31	3,020
40	97	31	3,030
41	111	31	3,040
42	125	31	3,180
43	139	31	3,257
44	153	31	3,214

A. Is there evidence of collinearity between the independent variables? Yes, temperature and length are collinear in that their correlation is quite high Yes, temperature and age of fish are collinear No, temperature and age have no correlation No, temperature and length have a low correlation Yes, Age and length have a high correlation None of the above B. What proportion of the variation in the response variable is explained by the regression? About 90 percent About 81 percent About 85 percent None of the above C. The F statistic indicates that: The regression, as a whole, is statistically significant More than half of the variation in Y is explained by the regression Age of fish is an important explanatory variable in the model Length of fish is an important explanatory variable in the model Water temperature is an important explanatory variable in the model None of the above D. The t-test of significance indicates that: The regression, as a whole, is statistically significant More than half of the variation in Y is explained by the regression Age of fish contributes information in the prediction of length of fish Length of fish contributes information in the prediction of age of fish Length of fish contributes information in the prediction of temperature E. The t-test of significance indicates that (same question but choose the correct answer): The regression, as a whole, is statistically significant More than half of the variation in Y is explained by the regression Length of fish is an important explanatory variable in the model Water temperature is an important explanatory variable in the model None of the above F. Assuming you ran the regression correctly, plot the residuals (against Y-hat). The plot shows that: The residuals appear to curve downwards, like a bowl facing down The residuals appear to curve upwards, like a bowl facing up (V shape) The residuals appear to be fanning out and are mostly spread out at the end The residuals appear random None of the above G. REGRESSION. Which of the following types of transformation may be appropriate given the shape of the residual plot? Logarithmic transformation in both Y and the X variables Quadratic transformation to correct for curvilinear relationship No transformation is necessary G. REGRESSION. This type of dataset is best described as a ____ and a residual problem common with this type of data is ___ Cross-sectional data; heteroscedasticity Time series data; heteroscedasticity Cross-sectional data; residual correlation Time series data; residual correlation Cross-sectional data; multicollinearity None of the above

Answer 1

a)

	Age	Temp	Length
Age	1
Temp	0	1
Length	0.879116	-0.18112	1

Yes, Age and length have a high correlation

b)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.897579068
R Square	0.805648183
Adjusted R Square	0.796167607
Standard Error	599.9975172
Observations	44
ANOVA
	df	SS	MS	F	Significance F
Regression	2	61184242.95	30592121.48	84.97881851	2.60655E-15
Residual	41	14759877.84	359997.0206
Total	43	75944120.8
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3904.266017	1149.044334	3.397837579	0.001522221	1583.723908	6224.808125
Age	26.24068177	2.055092802	12.76861159	7.11414E-16	22.09033765	30.39102588
Temp	-106.4136364	40.45182435	-2.630626382	0.011951331	-188.107753	-24.71951975

R^2 = 0.8056
hence about 81 %

c)
The regression, as a whole, is statistically significant
d)
Age of fish contributes information in the prediction of length of fish

e)
Water temperature is an important explanatory variable in the model