Q. 22 - Elementary Statistics

Question

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.

Step-by-Step Solution

Answer

(a) The error sum of the square is $1022.84$.

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

(c) The student-to-faculty ratio accounts for $26.1$ percent of the overall range in observed graduation rates.
(d) The regression equation isn't particularly useful for predicting.

Part (a) Step 1: Given information

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

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

SF_RATIO (x)	GRADUATE (y)	${(y-\overline{y})}^2$
16	45	42.25
20	55	12.25
17	70	342.25
19	50	2.25
22	47	20.25
17	46	30.25
17	50	2.25
17	66	210.25
10	26	650.25
18	60	72.25
	$\sum y_i=515$	$\sum {(y-\overline{y})}^2=1384.5$

Part (a)Step 3: Explanation

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:

Part (a) Step 4: Explanation

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:

SF_RATIO (x)	GRADUATE (y)	$\hat{y}=16.45+2.2026 \left(x\right)$	${(y-\overline{y})}^2$
16	45	48.866	6.937956
20	55	56.97	29.9209
17	70	50.892	0.369664
19	50	54.944	11.86144
22	47	61.022	90.66848
17	46	50.892	0.369664
17	50	50.892	0.369664
17	66	50.892	0.369664
10	26	36.71	218.7441
18	60	52.918	2.010724
	$\sum y_i=515$		${\sum (y-\overline{y})}^2=361.66$

Part (a) Step 5: Explanation

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 $1022.84$.

Part (b) Step 1: Given information

To obtain the coefficient of determination.

Part (b) Step 2: Explanation

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

$=\frac{361.66}{1384.5}$

$=0.261$
Asa result, the coefficient of determination is$0.261$.

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

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

Part (d) Sep 1: Given information

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

Part (d) Step 2: Explanation

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