Question

A student used multiple regression analysis to study how family spending (y) is influenced by income (x1), family size (x2), and additions to savings (x3). The variables y, x1, and x3 are measured in thousands of dollars. The following results were obtained.

Anova
	df	ss
regression	3	45.9634
residual	11	2.6218
Total

	coefficient	Standard error
intercept	0.0136
x1	0.7992	0.074
x2	0.2280	0.190
x3	-0.5796	0.920

Calculate the estimated regression equation for the relationship between the variables,coefficient of determination. What can you say about the strength of this relationship,Carry out a test to determine whether y is significantly related to the independent variables and Carry out a test to see if x3 and y are significantly related.

Use a 5% level of significance.

.

Answer 1

Sol:

Estimated regression eq uation is

y^=0.0136+0.7992*x1+0.2280*x2-0.5796*x3

,coefficient of determination=R sq=1-ss reg/ ss total

=1-(45.9634/48.5852)

= 0.9460371

=0.9460371*100

=94.60% avriation in y is expolained by model

R sq=0.9560

r=sqrt(0.9560)

= 0.9777525

There exists a strong positive relationship between y and indpeendent variable

Carry out a test to determine whether y is significantly related to the independent variables

Ho:beta1=beta2=beta3=0

Ha:atleast one of the betai is not =0

alpha=0.05

F=MS regressssion/MS residual

MS regresssion=SS regression/df

=45.9634/3

=15.32113

MS residual=Ss residual/df residual

=2.6218/11

=0.2383455

F=15.32113/0.2383455

F= 64.28118

p value in excel=

=F.DIST.RT( 64.28118,3,11)

=2.93051E-07

p value=0.000000293

p<0.05

Reject Ho

Accept Ha

Cocnlsuion:

There is suffcient statistical evidence at 5% level of significance to conclude

there is arelationship between y and independent variables that y is significantly related to the independent variables (x1,x2,x3)

Carry out a test to see if x3 and y are significantly related.

Ho:beta3=0

Ha:beta3 not =0

alpha=0.05

t stat=coeffcient/standard error

=-0.5796/0.920

=-0.63

df=n-k-1=15-3-1=11

=T.DIST.2T(0.63,11)

=0.54155995

p value>0.05

Fail to reject Ho

Accept Ho

There is no suffcient statistical evidence at 5% level of significance to conclude that y and x3 are related

x3 is not a significant variable in predicting y

-0.5796

0.920