23. Graduation Rates. Refer to Problem 21.
a. Compute the linear correlation coefficient, $r$.
b. Interpret your answer from part (a) in terms of the linear relationship between student-to-faculty ratio and graduation rate.
c. Discuss the graphical implications of the value of the linear correlation coefficient, $r$.
d. Use your answer from part (a) to obtain the coefficient of determination.

To compute the linear correlation coefficient, $r$of the function.

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 linear correlation coefficient is calculated as follows:

The correlation coefficient is calculated as follows:
$r=\frac{\sum x_i{y}_i-\frac{\sum x_i\left(\sum y_i\right)}{n}}{\sqrt{\left(\sum x_i^2-\frac{{\left(\sum x_i\right)}^2}{n}\right)\left(\sum y_i^2-\frac{{\left(\sum y_i\right)}^2}{n}\right)}}$

$=\frac{9088-8909.5}{\sqrt{(88.1)(1384.5)}}$

$=0.511$

As a result, the linear correlation coefficient is $r=0.511$.

To interpret the answer from part (a) in terms of the linear relationship between student-to-faculty ratio and graduation rate.

As a result from part (a) , the linear correlation coefficient is $r=0.511$.
The student-to-faculty ratio and graduation rate have a weekly positive relationship.

To discuss the graphical implications of the value of the linear correlation coefficient, $r$.

The scatterplot for the student-to-faculty ratio and graduation rate is presented below, created with MINITAB:

The data points are widely spread around the regression line, as seen in the scatter plot.
As a result, the student-to-faculty ratio and graduation rate have a weekly positive association.

To use the answer from part (a) to obtain the coefficient of determination.

As a result, from part (a) $r=0.511$
The coefficient of determination is calculated as:
$r^2=(0.511{)}^2$
$=0.261$
The coefficient of determination is $0.261$.