22. 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 (i.e. by the regression line).

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

To determine $SST,SSR$, and $SSE$by using the computing formulas.

In US colleges and universities, the graduation rate - the percentage of new freshmen who participate full-time and graduate within five years - and the factors that determine it have become a source of worry.
The following data on the student-to-faculty ratio (S/F ratio) and graduation rate came from a random sample of ten universities (Grad rate).

The following table shows how to compute SST for the original data:

Let, the sample size is $n=10$.
The mean of the graduate rates is calculated as:
$\overline{y}=\frac{\sum y_i}{n}$
$=\frac{515}{10}$

$=51.5$

The total sum of squares is determined as follows:

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

$=1384.5$

The regression output for the provided data using MINITAB is presented below:

The regression line can be seen in the output as follows:
$\hat{y}=16.4+2.03\left(x\right)$
The table for calculating $SSR$for the original data as follow:

The error sum of squares is expressed as follows:
$SSR=\sum (\hat{y}-\overline{y}{)}^2$
$=361.66$
The regression sum of squares is thus $361.66$. The total sum of squares is the regression sum plus the error sum of squares, as follows:
$\text{SST}=SSR+SSE$
$\Rightarrow SSE=SST-SSR$
$=1384.5-361.66$
$=1022.84$
As a result, the error sum of square is $1022.84$.

To obtain the coefficient of determination.

The coefficient of determination is calculated as follows:
$r^2=\frac{SSR}{SST}$

$=\frac{361.66}{1384.5}$

To obtain the percentage of the total variation in the observed graduation rates that is explained by student-to-faculty ratio.

The student-to-faculty ratio accounts for $26.1$ percent of the overall range in observed graduation rates.

To state how useful the regression equation appears to be for making predictions.

The regression equation isn't particularly useful for predicting from the given data.