Linear Statistical Models Quiz 1: 7 July 2024 (May 2024 term)

1. Consider the following R code which contains the information of marks of six students in Statistics and Mathematics subjects.

Based on the given information, answer the following questions.

(i) What will be the output of the command head(Marks, 2)? [2 Marks]

(ii) What will be the output of the command Marks[2,3]? [2 Marks]

(iii) Write an R code that can be used to extract the marks of fourth student in Stats and Maths. [2 Marks]

1. Let $X_1, X_2, X_3 \sim$ iid $N(0, 1)$ and $Y \sim \chi^2_n$. If each $X_i$ $(i = 1, 2, 3)$ is independent of $Y$, then which of the following options is(are) true? [2 Marks]

(a) $X_1^2 + X_2^2 + X_3^2 \sim t_3$

(b) $X_1^2 + X_2^2 + X_3^2 \sim \chi^2_3$

(c) $\dfrac{X_1}{\sqrt{\dfrac{Y}{n}}} \sim t_n$

(d) $\dfrac{X_2}{\sqrt{\dfrac{Y}{n}}} \sim t_{n-1}$

1. Consider a one way classification model and following table contains the observations, $y_{ij}$, of a data which has been divided into two populations (classes).

Based on the given information, answer the following questions.

(i). Calculate $\overline{Y}_{1o}, \overline{Y}_{2o}, \overline{Y}_{oo}$ and interpret your results. [4 Marks]

(ii). Find the value of sum of squares within group (SSW). [4 Marks]

(iii). Find the value of sum of squares between group (SSB). [2 Marks]

(iv). Calculate the value of total sum of squares (TSS). [4 Marks]

(v). Verify that TSS $=$ SSB $+$ SSW. [1 Mark]

1. Write the definition of “Least square estimation” and define all the notations properly. [3 Marks]

2. Consider a model:

$\begin{aligned} y_1 &= 5\beta_1 + 2\beta_2 + \epsilon_1 \;\; ; \\ y_2 &= 3\beta_1 - 2\beta_2 + \epsilon_2. \end{aligned}$

where, $\epsilon_i$'s $(i = 1, 2)$ are uncorrelated random variables with variance 1.

(i) Find the mathematical expression for $\|\vec{Y} - X\vec{\beta}\|_2$. [3 Marks]

Hint: If $z \in \mathbb{R}^n$, then $\|z\|_2 = \sqrt{z^T z} = \sqrt{\sum_{i=1}^{n} z_i^2}$.

(ii) Let $\vec{Y} = \begin{pmatrix} 10 \\ 6 \end{pmatrix}$, then which value of vector $\vec{\beta}$ among the following will minimize the expression $\|\vec{Y} - X\vec{\beta}\|_2$ ? [4 Marks]

(a) $\vec{\beta} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

(b) $\vec{\beta} = \begin{pmatrix} -1 \\ -1 \end{pmatrix}$

(c) $\vec{\beta} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

(d) Cannot determine.