4.3.3 Some Examples, 4.3.4 The Straight Line

Let $Y_{i} = \beta_{0}+\beta_{1} x_{i 1}+\cdots +\beta_{p-1} x_{i, p-1}+\varepsilon_{i}, i = 1,2, \ldots , n$, where the $\varepsilon_{i}$ are independent $N\left(0, \sigma^{2}\right)$. Prove that the $F$-statistic for testing the hypothesis $H: \beta_{q} = \beta_{q+1} = \cdots = \beta_{p-1} = 0(0 < q \leq p-1)$ is unchanged if a constant, c , say, is subtracted from each $Y_{i}$.

Let $Y_{i} = \beta_{0}+\beta_{1} x_{i}+\varepsilon_{i},(i = 1,2, \ldots , n)$, where the $\varepsilon_{i}$ are independent $N\left(0, \sigma^{2}\right)$.

Given that $\bar{x} = 0$, derive an $F$-statistic for testing the hypothesis $H$ : $\beta_{0} = \beta_{1}$ in Exercise No. 2 above. Show that it is equivalent to a certain $t$-test.

Let

and

where the $\varepsilon_{i}(i = 1,2,3)$ are independent $N\left(0, \sigma^{2}\right)$. Derive an $F$-statistic for testing the hypothesis $H: \theta_{1} = 2 \theta_{2}$.

Given $\mathbf{Y} = \theta +\varepsilon$, where $\varepsilon \sim N_{4}\left(\mathbf{0}, \sigma^{2} \mathbf{I}_{4}\right)$ and $\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4} = 0$, show that the $F$-statistic for testing $H: \theta_{1} = \theta_{3}$ is