Question

Consider the following data for a dependent variable y and two independent variables, x₁ and x₂.

x1	x2	y
30	13	95
46	10	108
25	18	113
50	16	179
40	5	95
51	20	176
74	7	170
36	12	117
59	13	142
77	16	211

Round your all answers to two decimal places. Enter negative values as negative numbers, if necessary.

a. Develop an estimated regression equation relating y to x₁.

ŷ =_________ +___________ x₁

Predict y if x₁ = 45.

ŷ = ____________

b. Develop an estimated regression equation relating y to x₂.

ŷ =__________ +____________ x₂

Predict y if x₂ = 15.

ŷ = ___________

c. Develop an estimated regression equation relating y to x₁ and x₂.

ŷ =________ +___________ x1________ +____________ x2

Predict y if x₁ = 45 and x₂ = 15.

ŷ = __________

Answer 1

a)

Regression Statistics
Multiple R	0.8090
R Square	0.6545
Adjusted R Square	0.6113
Standard Error	25.5286
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	9876.7	9876.7	15.16	0.0046
Residual	8	5213.7	651.7
Total	9	15090.4
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	47.4305	25.2577	1.8779	0.0972	-10.8139	105.6748
X	1.9092	0.4904	3.8930	0.0046	0.7783	3.0401

Ŷ =   47.43   +   1.91   *x

Predicted Y at X=   45   is                  
Ŷ =   47.4305   +   1.9092   *   45   =   133.35

b)

Regression Statistics
Multiple R	0.4273
R Square	0.1826
Adjusted R Square	0.0804
Standard Error	39.2673
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2755.0	2755.0	1.79	0.2181
Residual	8	12335.4	1541.9
Total	9	15090.4
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	92.5901	38.0028	2.4364	0.0408	4.9554	180.2248
X	3.6931	2.7628	1.3367	0.2181	-2.6780	10.0642

Ŷ =   92.59   +   3.69   *x

Predicted Y at X=   15   is                  
Ŷ =   92.5901   +   3.6931   *   15   =   147.99

c)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.957381
R Square	0.916579
Adjusted R Square	0.892744
Standard Error	13.41035
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	2	13831.54	6915.769	38.45567	0.000168
Residual	7	1258.862	179.8374
Total	9	15090.4
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-16.3966	19.00773	-0.86263	0.416915	-61.3427	28.54957	-61.3427	28.54957
x1	2.03232	0.258959	7.848043	0.000103	1.419979	2.64466	1.419979	2.64466
x2	4.447643	0.948435	4.689455	0.002236	2.204951	6.690336	2.204951	6.690336

Y^ = -16.40+2.03*X1 + 4.45*X2

Y^ = -16.40+2.03*45 + 4.45*15 = 141.70