4.9. A three-variable regression gave the following results: Mean sum of d.f. qu

Question

4.9. A three-variable regression gave the following results: Mean sum of d.f. quares (MSS) Sum of squares Source of variation Due to regression (ESS) 65,965 Due to residual (RSS) Total (TSS) 66,042 14 a. What is the sample size? b. What is the value of the RSS? c. What are the d.f. of the ESS and RSS? d. What is R2? And R2 e. Test the hypothesis that X2 and X3 have zero influence on Y. Which test do you use and why? f. From the preceding information, can you determine the individual contri- bution of X2 and X3 toward Y?

Explanation / Answer

We know for mutiple linear regression of p variables the degrees of freedom for the source of variations are given by

due to regression = p

due to residual=n-p-1

total =n-1

where n is number of observations on each of the 3 variables.

(a) total df =n-1=14

hence the sample size n=14+1=15

(b)RSS= TSS-ESS=77

(c)df of ESS = p=3

df of RSS =n-p-1=11

(d) R2 = (explained variability/total variability)= 1-(unexplained variability/total variability)

[

total variability= explained variability +unexplained variability

i.e. TSS=RSS+ESS ]

hence R2 = (ESS/TSS ) = 0.99

(R bar) 2 = adjusted R2 = 1- [{(1-R2)*(n-1)}/(n-p-1)] =1-[{(1-0.99)*(15-1)}/(15-3-1)]= 0.98

(e) our linear regression model of X on Y is

Y= a+b1X1 +b2X2+b3X3

where a is constant.

here we are to test the null hypothesis

H0 : b2 = b3 =0 against the alternative hypothesis

H1 : H0 is not true

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