Graduation Rates. Refer to Problem 21.

a. Determine SST. SSR. and SSE by using the computing formulas

b. Obtain the coefficient of determination.

c. Obtain the percentage of the total variation in the observed graduation rates that is explained by student-to-faculty ratio (ie.. by the regression line).

d. State how useful the regression equation appears to be for making predictions.

Graduation rate-the percentage of entering freshmen attending full time and graduating within 5 years and what influences it have become a concern in U.S colleges and universities. A random sample of 10 universities gave the following data on student-to-faculty ratio (S/F ratio) and graduation rate (Grad rate))

Table of computing SST for the original data as shown below:

The sample size, 10

By definition, Mean of the graduate rates is given by

$\overline{y}=\frac{{\displaystyle \sum _{}}{y}_i}{n}=\frac{515}{10}=51.5$

By definition, the total sum of squares is given by

$SST=\sum _{}{(y-\overline{y})}^2=\boxed{1384.5}$

Using MINITAB the regression output for the given data is shown below

From the output the regression line is

$\hat{y}=16.4+2.03(x)$

By definition, the error sum of squares is given by

SSR=$\sum _{}(y-\overline{y}){}^2=\boxed{361.66}$

Therefore, the regression sum of squares is 361.66

We know that the total sum of squares equals the regression sum of squares plus the error sum

of squares

SST= SSR+SSE

=>SSE=SST-SSR

=1384. 5-361.66

=1022. 84

Therefore, the error sum of square is 1022. 84

By definition, the coefficient of determination is given by

$r^2=\frac{SSR}{SST}=\frac{361.66}{1384.5}=\boxed{0.261}$

Therefore, the coefficient of determination is 0.261

The percentage of the total variation in the observed graduation rates that is explained by student-to-faculty ratio is 26.1%

The regression equation is not very useful for making prediction.