Question

Assume you have noted the following prices for books and the number of pages that each book contains. Book Pages (x) Price (y) A 500 $7.00 B 700 7.50 C 750 9.00 D 590 6.50 E 540 7.50 F 650 7.00 G 480 4.50 ​ a. Develop a least squares estimated regression line. b. Compute the coefficient of determination (r-squared). c. Compute the correlation coefficient between the price and the number of pages.

Answer 1

Answer:

a) $y=0.00991 x +1.042$  

b) $r^2 = 0.7503^2 = 0.563$

c) $r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503$  

Step-by-step explanation:

Data given

x: 500, 700, 750, 590 , 540, 650, 480

y: 7.00, 7.50 , 9.00, 6.5, 7.50 , 7.0, 4.50

Part a

We want to create a linear model like this :

$y = mx +b$

Wehre

$m=\frac{S_{xy}}{S_{xx}}$  

And:  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$  

With these we can find the sums:  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=2595100-\frac{4210^2}{7}=63085.714$  

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i){n}}=30095-\frac{4210*49}{7}=625$  

And the slope would be:  

$m=\frac{625}{63085.714}=0.00991$  

Nowe we can find the means for x and y like this:  

$\bar x= \frac{\sum x_i}{n}=\frac{4210}{7}=601.429$  

$\bar y= \frac{\sum y_i}{n}=\frac{49}{7}=7$  

And we can find the intercept using this:  

$b=\bar y -m \bar x=7-(0.00991*601.429)=1.042$  

And the line would be:

$y=0.00991 x +1.042$  

Part b

The correlation coefficient is given by:

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}$  

For our case we have this:

n=7 $\sum x = 4210, \sum y = 49, \sum xy = 30095, \sum x^2 =2595100, \sum y^2 =354$  

$r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503$  

The determination coefficient is given by:

$r^2 = 0.7503^2 = 0.563$

Part c

$r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503$