Respuesta :
Answer:
a) [tex]y=0.75 x -10.65[/tex]
b) [tex] y= 0.749*113 - 10.652=73.985\ approx 74.0[/tex]
Step-by-step explanation:
We assume that the data is this one:
x: 150 129 142 112 134 122 126 120
y: 88 96 106 80 98 63 95 64
Part a
We want to find a linear model [tex] y = mx+b[/tex]. Where:
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
Where:
[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]
So we can find the sums like this:
[tex]\sum_{i=1}^n x_i =1035[/tex]
[tex]\sum_{i=1}^n y_i =690[/tex]
[tex]\sum_{i=1}^n x^2_i =134965[/tex]
[tex]\sum_{i=1}^n y^2_i =61290[/tex]
[tex]\sum_{i=1}^n x_i y_i =90064[/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}=134965-\frac{1035^2}{8}=1061.875[/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}=90064-\frac{1035*690}{8}=795.25[/tex]
And the slope would be:
[tex]m=\frac{795.25}{1061.875}=0.749[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{1035}{8}=129.375[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{690}{8}=86.25[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=86.25-(0.749*129.375)=-10.652[/tex]
So the line would be given by:
[tex]y=0.75 x -10.65[/tex]
Part b
For this case w ejust need to replace x= 113 in the regression model and we got this:
[tex] y= 0.749*113 - 10.652=73.985\ approx 74.0[/tex]
