Question

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.

Short answer

(a) The linear correlation coefficient is $r=0.511$.

(b) The student-to-faculty ratio and graduation rate have a weekly positive relationship.

(c) The student-to-faculty ratio and graduation rate have a weekly positive association. The graphical implications of the value of the linear correlation coefficient as:

(d) The coefficient of determination is $0.261$.

Step-by-step solution

Part (a) Step 1: Given information

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

Part (a) Step 2: Explanation

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).

SF_RATIO	16	20	17	19	22	17	17	17	10	18
GRADUATE	45	55	70	50	47	46	50	66	26	60

The linear correlation coefficient is calculated as follows:

SF_RATIO (x)	GRADUATE (y)	$xy$	$x^2$	$y^2$
16	45	720	256	2025
20	55	1100	400	3025
17	70	1190	289	4900
19	50	950	361	2500
22	47	1034	484	2209
17	46	782	289	2116
17	50	850	289	2500
17	66	1122	289	4356
10	26	260	100	676
18	60	1080	324	3600
$\sum x_i=173$	$\sum y_i=515$	$\sum x_i{y}_i=9088$	$\sum {x_i}^2=3081$	$\sum {y_i}^2=27907$

Part (a) Step 3: Explanation

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 $r=0.511$.

Part (b) Step 1: Given information

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

Part (b) Step 2:Explanation

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.

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

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.

Part (d) Step 1: Given information

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

Part (d) Step 2: Explanation

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 $0.261$.