Math 6010 • Linear Models

(20 points) Researchers compare three classroom settings and measure a standardized reading score.
The observed scores are:
\[
\begin{aligned}
\text{Group 1: }&18, 21, 20 &&(n_{1} = 3),\\
\text{Group 2: }&23, 22, 24, 21 &&(n_{2} = 4),\\
\text{Group 3: }&26, 28, 27, 29, 25 &&(n_{3} = 5).
\end{aligned}
\]

Suppose the reading score for the \(i\)th individual in group \(j\) is modeled as
\[
Y_{ij} = \mu_{j} + \epsilon_{ij}, \qquad \epsilon_{ij} \sim \mathcal{N}(0, \sigma^{2}),
\]
with all errors independent.

Equivalently, this can be written in matrix form as
\[
Y = \mathbf{X}\beta + \epsilon.
\]
\begin{enumerate}
\item Write the model in matrix form by explicitly constructing the design matrix \(\mathbf{X}\), the parameter
vector \(\beta\), and the response vector \(Y\).
\item Express the null hypothesis ``all three group means are equal'' in the form
\[
H_{0}: A\beta = c,
\]
for explicit matrices \(A\) and \(c\).
\item You are told that the residual sum of squares under the full model is
\[
RSS = 38.4
\]
and under the reduced model (i.e. under \(H_{0}\)) it is
\[
RSS_{H_{0}} = 62.1.
\]
Compute the \(F\)-statistic for testing \(H_{0}\). State the degrees of freedom and give the conclusion
at level \(\alpha = 0.05\).
\item Could a \(t\)-test be used instead of an \(F\)-test here? Briefly explain.
\end{enumerate}