Respuesta :
Answer:
[tex]y=-2.95836 x +234.56159[/tex]
Step-by-step explanation:
We assume that th data is this one:
x: 50, 55, 50, 79, 44, 37, 70, 45, 49
y: 152, 48, 22, 35, 43, 171, 13, 185, 25
a) Compute the equation of the least-squares regression line. (Round your numerical values to five decimal places.)For this case we need to calculate the slope with the following formula:
[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 =50+ 55+ 50+ 79+ 44+ 37+ 70+ 45+ 49=479[/tex]
[tex]\sum_{i=1}^n y_i =152+ 48+ 22+ 35+ 43+ 171+ 13+ 185+ 25=694[/tex]
[tex]\sum_{i=1}^n x^2_i =50^2 + 55^2 + 50^2 + 79^2 + 44^2 + 37^2 + 70^2 + 45^2 + 49^2=26897[/tex]
[tex]\sum_{i=1}^n y^2_i =152^2 + 48^2 + 22^2 + 35^2 + 43^2 + 171^2 + 13^2 + 185^2 + 25^2=93226[/tex]
[tex]\sum_{i=1}^n x_i y_i =50*152+ 55*48+ 50*22+ 79*35+ 44*43+ 37*171+ 70*13+ 45*185+ 49*25=32784[/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}=26897-\frac{479^2}{9}=1403.556[/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)}=32784-\frac{479*694}{9}=-4152.22[/tex]
And the slope would be:
[tex]m=-\frac{-4152.222}{1403.556}=-2.95836[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{479}{9}=53.222[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{694}{9}=77.111[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=77.1111111-(-2.95836*53.22222222)=234.56159[/tex]
So the line would be given by:
[tex]y=-2.95836 x +234.56159[/tex]
