Answer:
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=438-\frac{44^2}{5}=50.8[/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}=415-\frac{44*42}{5}=45.4[/tex]
And the slope would be:
[tex]m=\frac{45.4}{50.8}=0.8937 \approx 0.894[/tex]
Now we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{44}{5}=8.8[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{42}{5}=8.4[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=8.4-(0.894*8.8)=0.535[/tex]
So the line would be given by:
[tex]y=0.894 x +0.535[/tex]
And the best option is:
A. y = 0.894x + 0.535
Step-by-step explanation:
We have the following dataset given
x: 5,6,9,10,14
y: 4,6,9,11,12
We want to find the least-squares line appropriate for this data given by this general expresion:
[tex] y = mx +b[/tex]
Where m is the slope and b the intercept
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 = 44[/tex]
[tex]\sum_{i=1}^n y_i =42[/tex]
[tex]\sum_{i=1}^n x^2_i =438[/tex]
[tex]\sum_{i=1}^n y^2_i =398[/tex]
[tex]\sum_{i=1}^n x_i y_i =415[/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}=438-\frac{44^2}{5}=50.8[/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}=415-\frac{44*42}{5}=45.4[/tex]
And the slope would be:
[tex]m=\frac{45.4}{50.8}=0.8937 \approx 0.894[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{44}{5}=8.8[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{42}{5}=8.4[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=8.4-(0.894*8.8)=0.535[/tex]
So the line would be given by:
[tex]y=0.894 x +0.535[/tex]
And the best option is:
A. y = 0.894x + 0.535
