Math 6010 • Linear Models

(20 points) Researchers study how a child's reading score \(Y\) depends on \emph{home reading time} \(x_{1}\)
(hours/week) and \emph{classroom support level} \(x_{2}\) (a small integer score). It is believed that extra home
reading time is more effective when classroom support is higher, so the mean is modeled with an
interaction between \(x_{1}\) and \(x_{2}\). To make the intercept interpretable, home reading time is centered
at \(1\) hour/week.

Consider the linear model
\[
Y_{i} = \beta_{0} + \beta_{1}(x_{1,i} - 1) + \beta_{2}(x_{1,i} \times x_{2,i}) + \epsilon_{i}, \qquad i = 1, \ldots, 5,
\]
where \(\epsilon_{i} \sim \mathcal{N}(0, \sigma^{2})\) are independent.
The observed data consist of:
\[
(x_{1}, x_{2}, Y) = (0, 1, 4), (1, 0, 3), (2, 1, 7), (1, 2, 6), (3, 1, 9).
\]

A least squares fit yields \(\hat{\beta} = (\hat{\beta}_{0}, \hat{\beta}_{1}, \hat{\beta}_{2})^{\top} = (2.10, 0.90, 1.20)^{\top}\) and \(RSS = 1.50\).
\begin{enumerate}
\item Write down the design matrix \(\mathbf{X}\) for this regression model and the parameter vector \(\beta =
(\beta_{0}, \beta_{1}, \beta_{2})^{\top}\). Identify the estimated variance \(S^{2}\).
\item We wish to test the joint null hypothesis that there is \emph{no effect} of the predictors beyond the
baseline,
\[
H_{0}: \beta_{1} = \beta_{2} = 0.
\]
Express this hypothesis in matrix form \(A\beta = 0\) for an appropriate matrix \(A\), and state \(q =
\operatorname{rank}(A)\).
\item Write down the \(F\)-statistic for testing \(H_{0}\) and state the distribution of \(F\) under \(H_{0}\). (You do
not need to evaluate the formula.)
\item Using Scheff\'e's method, construct \emph{simultaneous} 95\% confidence intervals for \(\beta_{0}, \beta_{1}, \beta_{2}\). You
may express these intervals in terms of \((\mathbf{X}^{\top}\mathbf{X})^{-1}\), but plug in all other quantities.
\end{enumerate}