Question

The following equation was used:
?[?]=?0+?1?1+?2?2+?3?3E[y]=β0+β1x1+β2x2+β3x3
where
?1={10if level 2o.w.x1={1if level 20o.w. ?2={10if level 3o.w.x2={1if level 30o.w. ?3={10if level 4o.w.x3={1if level 40o.w.

The model was fitted to n=30 data points, and the following result was obtained:
?̂ =10.2−4?1+12?2+2?3y^=10.2−4x1+12x2+2x3

Use the least squares equation above to find the estimated mean for each level of the qualitative independent variable.
Estimated mean for level 1:
Estimated mean for level 2:
Estimated mean for level 3:
Estimated mean for level 4:

FOR WEBWORK

Answer 1

For level 1, x1 = x2 = x3 = 0

Estimated mean for level 1:

y^ = 10.2 - 4 * 0 +12 * 0 + 2 * 0 = 10.2

For level 2, x1 = 1 , x2 = x3 = 0

Estimated mean for level 2:

y^ = 10.2 - 4 * 1 +12 * 0 + 2 * 0 = 6.2

For level 3, x2 = 1 , x1 = x3 = 0

Estimated mean for level 3:

y^ = 10.2 - 4 * 0 +12 * 1 + 2 * 0 = 22.2

For level 4,   x3 = 1 , x1 = x2 = 0

Estimated mean for level 4:

y^ = 10.2 - 4 * 0 +12 * 0 + 2 * 1 = 12.2