Question

A set of (x,y) paired data values yields the following results: the x values have a sample mean of 12, the y values have a sample mean of 38, the y values have a sample standard deviation of 4, the x values have a sample standard deviation of 3, and the correlation coefficient between x and y is .76. Compute the least squares regression line for the data, and then predict what the y value should be if x is 4.

Answer 1

Least square regression:

y = 1.0136x + 25.8404

Value of y if x is 4:

y = 29.8936

Explanation:

Given:

x' = 12

y' = 38

`\sigma y = 4`

`\sigma x = 3`

coefficient between x and y, r = 0.76

Take the least square regression as:

y - y' = byx (x - x')

Let's find byx:

`byx = r ((\sigma y)/(\sigma x))`

`byx = 0.76 ((4)/(3)) = 1.0133`

We'll now compute the least squares regression line for the data, and then predict what the y value should be if x is 4.

Take the regression equation:

y - y' = byx (x - x')

Substitute figures:

y - 38 = 1.0133 (x - 12)

Expand the equation:

y - 38 = 1.0133x - 12.1596

y = 1.0133x - 12.1596 + 38

y = 1.0136x + 25.8404

The y value when x= 4:

y - 38 = 1.0133(4) + 25.8404

y - 38 = 4.0532 + 25.8404

y = 29.8936

Therefore,

Least square regression:

y = 1.0136x + 25.8404

Value of y if x is 4:

y = 29.8936