Question 1

1> Math_marks = c(70, 85, 92, 60, 55, 60)
2> Stats_marks = c(80, 70, 80, 90, 75, 65)
3> Marks = data.frame(Math_marks, Stats_marks)
4> Marks
5  Math_marks Stats_marks
61         70          80
72         85          70
83         92          80
94         60          90
105         55          75
116         60          65

1. Consider the following R code which contains the information of marks of six students in Statistics and Mathematics subjects. > Math_marks = c(70, 85, 92, 60, 55, 60) > Stats_marks = c(80, 70, 80, 90, 75, 65) > Marks = data.frame(Math_marks, Stats_marks) > Marks Math_marks Stats_marks 1 70 80 2 85 70 3 92 80 4 60 90 5 55 75 6 60 65 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] 2. 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) $\dfrac{X_2}{\sqrt{\dfrac{Y}{n}}} \sim t_{n-1}$ Scatter plot of Calories Burned vs Time Exercised (hours), with questions on it and on one way classification statements 5. 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). | Population 1 | Population 2 | |---|---| | 4 | 5 | | 5 | 5 | | 4 | 4 | | 7 | 6 | 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] 6. Write the definition of “Least square estimation” and define all the notations properly. [3 Marks] 7. 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.