In a small-scale regression study, five observations on $Y$ were
obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that
$\sigma=0.6,B_0=5,B_1=3$a. What are the expected values off MSR and MSE here?
b. For determining whether or not a regression relation exists, would
it has been better or worse to have made the five observations at $X
= 6,7, 8, 9$, and $10$? Why? Would the same answer apply if the principal purpose were to estimate the mean response for $X = 8$?
Discuss.
$$Y_i=B_0+B_1X_i+\epsilon_i$$
$$\hat{Y_i}=\hat{B_0}+\hat{B_1}X_i$$
$$MSR=\sum(\hat{Y_i}-\overline{Y})^2$$
$$MSE=\frac{\sum (Y_i-\hat{Y_i)}^2}{n-2}=\frac{\sum(B_0+B_1X_i+\epsilon_i-\hat{B_0}-\hat{B_1}X_i)^2}{n-2}$$
I'm still doesn't understand what they want, they want
$$E(MSE);E(MSR)$$?
What do they mean by expected values?
Best Answer
$$Y_i=B_0+B_1X_i+\epsilon_i$$ $$\hat{Y_i}=\hat{B_0}+\hat{B_1}X_i$$
a) $$E[MSE]=E[\frac{\sum(Y_i-\hat{Y_i})^2}{n-2}]=\sigma^2=0.6^2$$ $$E[MSR]=E[\sum(\hat{Y_i}-\overline{Y})^2]=\sigma^2+B_1\sum(X_i-\overline{X})^2=1026.36$$
b) $$\sigma(\hat{B_1})=\sqrt{\frac{\sigma^2}{\sum(X_i-\overline{X})^2}}=\frac{0.6}{\sqrt{\sum(X_i-\overline{X})^2}}$$ for the case where $X=(1,4,10,11,14)$ we have that $\sigma(\hat{B_1})=0.05619515$ and for the case where $X=(6,7,8,9,10)$ $\sigma(B_1)=0.1897367$, then the first set is better I think.
But why I need to look $\sigma(\hat{B_1})$?
Is there any difference if it were estimating the mean response for X = 8?