Question

A researcher would like to predict the dependent variable Y from the two independent variables X1 and X2 for a sample of N=16 subjects. Use multiple linear regression to calculate the coefficient of multiple determination and test the significance of the overall regression model. Use a significance level α=0.05.

X1             X2             Y

48	42.3	47.1
36.3	58.7	65.4
43.4	40.2	63.6
49.5	37.9	45.6
45.5	37.2	50.8
40.6	64.7	42.4
42.5	46.7	63.1
42.7	40	35.8
55.8	10.6	52.1
40.9	63	60.3
39.6	56.5	44
43.5	45.1	61.2
39	68.8	67.2
50.4	43.7	40.6
46.1	42.6	58
55.2	19.1	49.1

SSreg=
SSres=
R2=
F=
P-value =

What is your decision for the hypothesis test?

• Reject the null hypothesis, H0:β1=β2=0

• Fail to reject H0

What is your final conclusion?

• The evidence supports the claim that one or more of the regression coefficients is non-zero
• The evidence supports the claim that all of the regression coefficients are zero
• There is insufficient evidence to support the claim that at least one of the regression coefficients is non-zero
• There is insufficient evidence to support the claim that all of the regression coefficients are equal to zero

Answer 1

Applying regression on above data:

Regression Statistics
Multiple R	0.3911
R Square	0.1530
Adjusted R Square	0.0227
Standard Error	9.7816
Observations	16
ANOVA
	df	SS	MS	F	Significance F
Regression	2	224.6738	112.3369	1.1741	0.3398
Residual	13	1243.8355	95.6797
Total	15	1468.5094
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	110.5744	57.4787	1.9237	0.0766	-13.6007	234.7496	-13.6007	234.7496
x1	-1.0988	0.9613	-1.1430	0.2737	-3.1756	0.9781	-3.1756	0.9781
x2	-0.1853	0.3474	-0.5333	0.6028	-0.9357	0.5652	-0.9357	0.5652

from above"

SSreg =224.6738

SSres=1243.8355

R2= 0.1530

F =1.1741

p value =0.3398

Fail to reject H0

There is insufficient evidence to support the claim that at least one of the regression coefficients is non-zero