| course |
course_year |
question_number |
tags |
title |
year |
Statistics |
IB |
75 |
|
Paper 1, Section II, H |
2013 |
Consider the general linear model $Y=X \theta+\epsilon$ where $X$ is a known $n \times p$ matrix, $\theta$ is an unknown $p \times 1$ vector of parameters, and $\epsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with unknown variance $\sigma^{2}$. Assume the $p \times p$ matrix $X^{T} X$ is invertible. Let
$$\begin{aligned}
\hat{\theta} &=\left(X^{T} X\right)^{-1} X^{T} Y \\
\hat{\epsilon} &=Y-X \hat{\theta}
\end{aligned}$$
What are the distributions of $\hat{\theta}$ and $\hat{\epsilon}$ ? Show that $\hat{\theta}$ and $\hat{\epsilon}$ are uncorrelated.
Four apple trees stand in a $2 \times 2$ rectangular grid. The annual yield of the tree at coordinate $(i, j)$ conforms to the model
$$y_{i j}=\alpha_{i}+\beta x_{i j}+\epsilon_{i j}, \quad i, j \in{1,2},$$
where $x_{i j}$ is the amount of fertilizer applied to tree $(i, j), \alpha_{1}, \alpha_{2}$ may differ because of varying soil across rows, and the $\epsilon_{i j}$ are $N\left(0, \sigma^{2}\right)$ random variables that are independent of one another and from year to year. The following two possible experiments are to be compared:
$$\mathrm{I}:\left(x_{i j}\right)=\left(\begin{array}{cc}
0 & 1 \\
2 & 3
\end{array}\right) \quad \text { and } \quad \mathrm{II}:\left(x_{i j}\right)=\left(\begin{array}{cc}
0 & 2 \\
3 & 1
\end{array}\right) \text {. }$$
Represent these as general linear models, with $\theta=\left(\alpha_{1}, \alpha_{2}, \beta\right)$. Compare the variances of estimates of $\beta$ under I and II.
With II the following yields are observed:
$$\left(y_{i j}\right)=\left(\begin{array}{ll}
100 & 300 \\
600 & 400
\end{array}\right)$$
Forecast the total yield that will be obtained next year if no fertilizer is used. What is the $95 %$ predictive interval for this yield?