Respuesta :
Answer:
a) [tex]y=0.00991 x +1.042[/tex]
b) [tex]r^2 = 0.7503^2 = 0.563[/tex]
c) [tex]r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503[/tex]
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 :
[tex] y = mx +b[/tex]
Wehre
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
And:
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]
With these we can find the sums:
[tex]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[/tex]
[tex]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[/tex]
And the slope would be:
[tex]m=\frac{625}{63085.714}=0.00991[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{4210}{7}=601.429[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{49}{7}=7[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=7-(0.00991*601.429)=1.042[/tex]
And the line would be:
[tex]y=0.00991 x +1.042[/tex]
Part b
The correlation coefficient is given by:
[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]
For our case we have this:
n=7 [tex] \sum x = 4210, \sum y = 49, \sum xy = 30095, \sum x^2 =2595100, \sum y^2 =354[/tex]
[tex]r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503[/tex]
The determination coefficient is given by:
[tex]r^2 = 0.7503^2 = 0.563[/tex]
Part c
[tex]r=\frac{7(30095)-(4210)(49)}{\sqrt{[7(2595100) -(4210)^2][7(354) -(49)^2]}}=0.7503[/tex]
