12. A professor obtains SAT scores and freshman grade point averages (GPAs) for

Question

12. A professor obtains SAT scores and freshman grade point averages (GPAs) for a group of n = 15 college students. The SAT scores have a mean of M = 580 with SS 22,400, and the GPAs have a mean of 3.10 with SS 1.26, and SP- 84 a. Find the regression equation for predicting GPA from SAT scores. b. What percentage of the variance in GPAs is accounted for by the regression equation? (Compute the correlation, r then find r2.) c. Does the regression equation account for a significant portion of the variance in GPA? Use ? = .05 to evaluate the F-ratio.

Explanation / Answer

a)

here slope b=SP/SSx =84/22400=0.00375

and intercept a =My-b*Mx=3.10-0.00375*580=0.925

therefore regression equation: Y=0.925+0.00375*x

b)

correlation r =SP/sqrt(SSx*SSy)=84/sqrt(22400*1.26)=0.5

% of variation that can be accounted by regression equation =r2*100 =25.0%

c)

here SST=SSy=1.26

SSR =r2*1.26=0.315

SSE =SST-SSR =0.945

degree of freedom of regression =df(regression)=independent variables=p =1

df(error)=n-p-1=13

hence test statsitic F =(SSR/df(regression))/(SSE/df(error))=4.333

at (1,13) degree of freedom and 0.05 l;evel critical value of F =4.667

as test statistic 4.333 is not in critical region we can not reject null hypothesis

we do not have evidence to conclude at 0.05 level that regression equation account for a signfiicant portion of the variance in GPA

source SS df MS F regression 0.315 1 0.315 4.333 error 0.945 13 0.073 total 1.26 14

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