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Sagot :
[tex]a) $y=0.00991 x+1.042$b) $r^2=0.7503^2=0.563$\\C) $r=\frac{7(30095)-(4210)(49)}{\sqrt{\left[7(2595100)-(4210)^2\right]\left[7(354)-(49)^2\right]}}=0.7503$[/tex]
x: 500, 700, 750, 590 , 540, 650, 480
y: 7.00, 7.50 , 9.00, 6.5, 7.50 , 7.0, 4.50
We want to create a linear model like this :
[tex]$y=m x+b$[/tex]
Where
[tex]$m=\frac{S_{x y}}{S_{x x}}$[/tex]
And:
[tex]$\begin{aligned}& S_{x y}=\sum_{i=1}^n x_i y_i-\frac{\left(\sum_{i=1}^n x_i\right)\left(\sum_{i=1}^n y_i\right)}{n} \\& S_{x x}=\sum_{i=1}^n x_i^2-\frac{\left(\sum_{i=1}^n x_i\right)^2}{n}\end{aligned}$[/tex]
With these we can find the sums:
[tex]$\begin{aligned}& S_{x x}=\sum_{i=1}^n x_i^2-\frac{\left(\sum_{i=1}^n x_i\right)^2}{n}=2595100-\frac{4210^2}{7}=63085.714 \\& S_{x y}=\sum_{i=1}^n x_i y_i-\frac{\left(\sum_{i=1}^n x_i\right)\left(\sum_{i=1}^n y_i\right) n}{=} 30095-\frac{4210 * 49}{7}=625\end{aligned}$[/tex]
And the slope would be:
[tex]m=\frac{625}{63085.714}=0.00991[/tex]
Now we can find the means for x and y like this:
[tex]$\begin{aligned}& \bar{x}=\frac{\sum x_i}{n}=\frac{4210}{7}=601.429 \\& \bar{y}=\frac{\sum y_i}{n}=\frac{49}{7}=7\end{aligned}$[/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\left(\sum x y\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n \sum x^2-\left(\sum x\right)^2\right]\left[n \sum y^2-\left(\sum y\right)^2\right]}}[/tex]
For our case we have this:
[tex]$\begin{aligned}& \mathrm{n}=7 \sum x=4210, \sum y=49, \sum x y=30095, \sum x^2=2595100, \sum y^2=354 \\& r=\frac{7(30095)-(4210)(49)}{\sqrt{\left[7(2595100)-(4210)^2\right]\left[7(354)-(49)^2\right]}}=0.7503\end{aligned}$[/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{\left[7(2595100)-(4210)^2\right]\left[7(354)-(49)^2\right]}}=0.7503[/tex]
Learn more about regression line to visit this link
https://brainly.com/question/7656407
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