Respuesta :
Answer:
[tex]m=\frac{332}{182}=1.824[/tex]
[tex]b=\bar y -m \bar x=43.615-(1.824*21)=5.311[/tex]
So the line would be given by:
[tex]y=1.824 x +5.311[/tex]
Step-by-step explanation:
We assume that the data is this one:
x: 15 16 17 18 19 20 21 22 23 24 25 26 27
y: 33 34 33 36 36 47 52 51 41 50 49 50 55
Find the least-squares line appropriate for this data.
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 =273[/tex]
[tex]\sum_{i=1}^n y_i =567[/tex]
[tex]\sum_{i=1}^n x^2_i =5915[/tex]
[tex]\sum_{i=1}^n y^2_i =25547[/tex]
[tex]\sum_{i=1}^n x_i y_i =12239[/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}=5915-\frac{273^2}{13}=182[/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}=12239-\frac{273*567}{13}=332[/tex]
And the slope would be:
[tex]m=\frac{332}{182}=1.824[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{273}{13}=21[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{567}{13}=43.615[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=43.615-(1.824*21)=5.311[/tex]
So the line would be given by:
[tex]y=1.824 x +5.311[/tex]
