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Nat2105 [25]
3 years ago
14

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
Business
1 answer:
svetoff [14.1K]3 years ago
5 0

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

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