Answer to Question #86958 in Statistics and Probability for Willy

Question

a) Calculate the regression equation of X on Y (5 mks)
X
1
2
3
4
5

Y
2
5
3
8
7

b) Explain the assumptions of the linear regression model (5Mks)

Answer 1

$$\begin{matrix} X & Y & XY & X^2 & Y^2 \\ 1 & 2 & 2 & 1 & 4 \\ 2 & 5 & 10 & 4 & 25 \\ 3 & 3 & 9 & 9 & 9 \\ 4 & 8 & 32 & 16 & 64 \\ 5 & 7 & 35 & 25 & 49 \\ \Sigma(X)=15 & \Sigma(Y)=25 & \Sigma(XY)=88 & \Sigma(X^2)=55 & \Sigma(Y^2)=151 \\ \end{matrix}$$ $$n=5$$

$$a=\frac{(\Sigma(X))( \Sigma(Y^2))-(\Sigma(Y))( \Sigma(XY))}{n (\Sigma(Y^2))-(\Sigma(Y))^2}$$

$$b=\frac{n(\Sigma(XY))-(\Sigma(Y))( \Sigma(X))}{n (\Sigma(Y^2))-(\Sigma(Y))^2}$$

$$a={15(151)-25(88) \over 5(151)-(25)^2}=0.5$$

$$b={5(88)-15(25) \over 5(151)-(25)^2}=0.5$$

The equation is

$$X=0.5Y+0.5$$

CLASSICAL LINEAR REGRESSION MODEL ASSUMPTIONS

1. A linear regression exists between the dependent variable and the independent variable.

2. The independent variable is not random.

3. The expected value of the error term is 0.

This assumption says that, on average, we expect the impact of all left-out factors in our model to be zero.

4. The variance for the error term is the same for all observations. (homoscedasticity; the complementary concept is called heteroscedasticity).

5. The error term is normally distributed.

6. The error term is uncorrelated across observations.