Hat Matrix and Residuals

(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 \[ Y = \mathbf {X}\beta + \epsilon . \]

1. Write the model in matrix form by explicitly constructing the design matrix \(\mathbf {X}\), the parameter vector \(\beta \), and the response vector \(Y\).
2. Express the null hypothesis “all three group means are equal” in the form \[ H_{0}: A\beta = c, \] for explicit matrices \(A\) and \(c\).
3. 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 \(\alpha = 0.05\).
4. Could a \(t\)-test be used instead of an \(F\)-test here? Briefly explain.