Q. 23 - Elementary Statistics

Question

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 past (a) to obtain the coefficient of determination .

Step-by-Step Solution

Answer

A) :  the linear correlation coefficient, r=0.511 

B) : The relationship between variables student to-faculty ratio and graduation rate is weekly positive.

C) : the relationship between variables student-to-faculty ratio and graduation rate is weekly positive. 

D) :  the coefficient of determination is $r^2 = {\left(0.511\right)}^2$

Step 1. Given

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))

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

Step 1. Part ( a )

The following is the table for obtaining the linear correlation coefficient.

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
$\sum _{}{x}_i= 173$	$\sum _{}{y}_i= 515$	$\sum _{}{x}_i{y}_i= 9088$	$\sum _{}{x}^2= 3081$	$\sum _{}{y}^2= 27907$

By definition, the correlation coefficient is given by

$$\begin{gathered} r = \frac{{\displaystyle \sum _{}}{x}_i{y}_i - {\displaystyle \frac{{\displaystyle \sum _{}}{x}_i\left({\displaystyle \sum _{}}{y}_i\right)}{n}}}{\sqrt{\left({\displaystyle \sum _{}}{x^2}_{i }-{\displaystyle \frac{{\displaystyle \underset{}{(\sum }}{x}_i{)}^2}{n}}\right)\left({\displaystyle \sum _{}}{y^2}_i - {\displaystyle \frac{{\left({\displaystyle \sum _{}}{y}_i\right)}^2}{n}}\right)}} \\ = \frac{9088-{\displaystyle \frac{\left(173\right)\left(515\right)}{10}}}{\sqrt{\left(3081-{\displaystyle \frac{{\left(173\right)}^2}{10}}\right)\left(27907 - {\displaystyle \frac{{\left(515\right)}^2}{10}}\right)}} \\ =\frac{9088 - 8909.5}{\sqrt{(88.1)(1384.5) }} = \boxed{0.511} \end{gathered}$$

Therefore, the linear correlation coefficient, r=0.511

Step 3. Part ( b )

From part (a) r=0.511

The relationship between variables student to-faculty ratio and graduation rate is weekly positive.

Step 4. Part ( c )

Using MINITAB the scatterplot for the student-to-faculty ratio and graduation rate is shown below

From scatter plot we can see that the data point are widely scattered about the regression line. So the relationship between variables student-to-faculty ratio and graduation rate is weekly positive.

Step 5. Part ( d )

From part (a) r=0.511

By definition, the coefficient of determination is

$r^2 = {\left(0.511\right)}^2$