CSIR NET Maths previous year question papers

Let $X_{1}, \dots, X_{n}$ be independent and identically distributed $U(0, \theta), \theta > 0$ random variables. Define $X_{(n)} = \max${$X_{1}, \dots, X_{n}$} and $X_{(1)} = \min${$X_{1}, \dots, X_{n}$}. Which of the following statements are true?

Let $X_{1}, \dots, X_{n} (n \ge 3)$ be a random sample from a distribution having probability density function f(x | $\theta) = \theta{}e^(-\theta{}x)$ if x > 0, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Let $T_{n} = (1/n)\sum_{i}$₌$_{1}^{n}X_{i}$. Which of the following statements are true?

Consider a six faced die whose i-th face is marked with i dots, i = 1, 2, …, 6. In a single random throw of the die, let $p_{i}$ denote the probability that the obtained upper face has i dots, i = 1, 2, …, 6. The die is rolled 240 times independently and the following result is obtained — face observed 1, 2, 3, 4, 5, 6 with frequencies 40, 55, 40, 25, 35, 45 respectively. Suppose we want to test $H_{0} : p_{i} = 1/6$ for i = 1, 2, …, 6; against $H_{1} : p_{i} \ne 1/6$ for at least one i. It is given that $\chi^{2}_{5};_{0}._{05} = 11.07, \chi^{2}_{6};_{0}._{05} = 12.59, \chi^{2}_{5};_{0}._{01} = 15.09, \chi^{2}_{6};_{0}._{01} = 16.81$. Based on the asymptotic goodness of fit $\chi^{2}$ test for testing $H_{0}$ against $H_{1}$, which of the following statements are true?

Observations on the shear strength of concrete from 5 randomly selected structures are given below — structure 1, 2, 3, 4, 5 with shear strength 1718.4, 1787.4, 2562.3, 2356.9, 2153.2 respectively. The null hypothesis $H_{0}$ that the median shear strength is 2000 units is tested against the alternative hypothesis $H_{1}$ that the median shear strength is greater than 2000 units at 5% level of significance. Which of the following statements are true?

Consider the following ANOVA table for a randomized block design — Treatments: sum of squares 48, degrees of freedom 4, mean squares 12, F calculated $\beta$; Blocks: sum of squares 72, degrees of freedom 3, mean squares 24, F calculated 12; Error: sum of squares $\alpha$, degrees of freedom m, mean squares $\gamma$; Total: sum of squares 144, degrees of freedom 19. Which of the following statements are true?