2. Short Answer: Give a brief answer, explanation, and/or mathematical derivation to the questions
below.
(a) Suppose it is observed that, for a set of data points {(Xi, Yi)} which are assumed to satisfy a
simple linear regression model, the sample mean of Yi is greater than the absolute value of the
sample mean of Xi, but the sample variance of the dependent variable Yi is less than the sample
variance of the independent variable Xi that is, $\bar{Y} > |\bar{X}|$ and $s_Y^2 < s_X^2$. What, if anything,
does this imply about the absolute value of the least-squares slope coefficient estimate $\hat{\beta}$? How
about the sign of the intercept term $\hat{\alpha}$?
(b) Suppose that, instead of fitting the regression model $Y_i = \beta_0 + \beta_1X_i + U_i$ by least squares, you
instead fit the model $Y_i = \gamma_0 + \gamma_1Z_i + U_i$, where $Z_i = 3 + 10X_i$. How are the least squares
estimators $\hat{\gamma}_0$ and $\hat{\gamma}_1$ for this second model related to the corresponding LS estimators $\beta_0$ and
$\hat{\beta}_1$ of the first model? [Hint: figure out how $\bar{Z}$, $\sigma_Z^2$, and $\sigma_{YZ}$ are related to $\bar{X}$, $\sigma_X$, and $\sigma_{YX}$.]
