Question

On the first statistics exam, the coefficient of determination between the hours studied and the grade earned was 80%. The standard error of estimate was 10. There were 20 students in the class. Develop an ANOVA table for the regression analysis of hours studied as a predictor of the grade earned on the first statistics exam.

Answer 1

Answer:

$\left[\begin{array}{ccccc}&DF&SS&MS&F\\Regression&1&7200&7200&72\\Error&18&1800&100\\total&19&900\end{array}\right]$

Explanation:

Sample size, n=20

Degrees of freedom is 1

Number of degrees of freedom for error is n-2 hence 20-2=18

Total number of degrees of freedom is 18+1=19

Standard error estimate is $s_{y-x}=\sqrt {\frac {SSE}{n-2}}$

Here, $SSE=(n-2)s_{y-x}^{2}=(20-2)(10)^{2}=1800$

Coefficient of determination $r^{2}=\frac {SSE}{SS total}$

Here, $SSR=r^{2}(SSR+SSE)$

$SSR=\frac {r^{2}}{1-r^{2}} SSE=\frac {0.8}{1-0.8}(1800)=7200$

The total sum of squares is

SS total=SSR+SSE=7200+1800=9000

MSR=SSR=7200

$MSE=\frac {SSE}{n-2}=\frac {1800}{20-2}=100$

F value is given by

$F=\frac {MSR}{MSE}=\frac {7200}{100}=100$

The ANOVA table is then  

$\left[\begin{array}{ccccc}&DF&SS&MS&F\\Regression&1&7200&7200&72\\Error&18&1800&100\\total&19&900\end{array}\right]$