Question

Below are a set of heights (in inches) and GPA scores for a sample of 6 students. Height, GPA 60, 4.0 55, 3.2 62, 3.7 55, 3.9 49, 2.4 61, 2.7 Find the equation of the regression line to predict GPA from height by hand

Answer 1

Answer:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=19616-\frac{342^2}{6}=122$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=1142.2-\frac{342*19.9}{6}=7.9$

And the slope would be:

$m=\frac{7.9}{122}=0.0648$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{342}{6}=57$

$\bar y= \frac{\sum y_i}{n}=\frac{19.9}{6}=3.317$

And we can find the intercept using this:

$b=\bar y -m \bar x=3.317-(0.0648*57)=-0.377$

So the line would be given by:

$y=0.0648 x -0.377$

Explanation:

The data given is:

x: 60, 55, 62, 55,49, 61

y: 4.0, 3.2, 3.7, 3.9, 2.4, 2.7

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i = 342$

$\sum_{i=1}^n y_i =19.9$

$\sum_{i=1}^n x^2_i =19616$

$\sum_{i=1}^n y^2_i =68.19$

$\sum_{i=1}^n x_i y_i =1142.2$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=19616-\frac{342^2}{6}=122$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=1142.2-\frac{342*19.9}{6}=7.9$

And the slope would be:

$m=\frac{7.9}{122}=0.0648$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{342}{6}=57$

$\bar y= \frac{\sum y_i}{n}=\frac{19.9}{6}=3.317$

And we can find the intercept using this:

$b=\bar y -m \bar x=3.317-(0.0648*57)=-0.377$

So the line would be given by:

$y=0.0648 x -0.377$