CSIR NET Maths previous year question papers

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a continuous distribution with the common probability density function f(x|$\theta) = 2^\theta \theta /$ $x^{\theta+1}$ if x > 2, and 0 otherwise, where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y > \chi^{2}_{m,\beta})$ $= \beta$, where Y ~ $\chi^{2}_m$. For testing $H_{0} : \theta = 1$ against $H_{1} : \theta > 1, a$ uniformly most powerful test of size $\alpha, 0 < \alpha < 1$, will reject $H_{0}$ if

Let $X_{1}$ and $X_{2}$ be a random sample from Uniform$[0,\theta]$ distribution, where $\theta > 0$. For testing the hypothesis $H_{0}: \theta=1$ against $H_{1}: \theta=2$, consider a test which rejects $H_{0}$ if $X_{1}+X_{2} > 4/5$. Then, the probability of type-I error is

Let X be a random sample of size one from the probability density function f(x|$\theta) =$ $\theta{}e^{-\theta(x-1)}$ if x > 1, and 0 otherwise, where $\theta (> 0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_{0} : \theta = 1$ against the alternative hypothesis $H_{1} : \theta \ne 1$, based on the observed value x of X. Then, which of the following statements are true?

Let $(X_{1}, X_{2}, \dots, X_{7})$ and $(Y_{1}, Y_{2}, \dots, Y_{9})$ be two independent random samples from the continuous distribution functions $F(x - \mu)$ and $F(x - \theta)$, respectively, where $F, \mu$ and $\theta$ are all unknown. Further, let $\mu$ be the unique median of $F(x - \mu)$ and $\theta$ be the unique median of $F(x - \theta)$. Let $R_{i}$ be the rank of $Y_{i}$ in the combined sample, i = 1, 2, …, 9. For testing $H_{0} : \mu = \theta$ against $H_{1} : \mu > \theta$, the test statistic $T = \sum_{i}$₌$_{1}^{9} R_{i}$ is proposed. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{15}$ be a random sample from an Exponential distribution with the probability density function f(x∣$\sigma) = (1/\sigma)\exp(-x/\sigma)$ if x>0, 0 elsewhere, where the unknown parameter $\sigma$ is positive. Let ${\bar{X}} = (1/15)\sum_{1}^{15}X_{i}$. Suppose that $\varphi$ denotes the likelihood ratio test for testing $H_{0}: \sigma\le1$ against $H_{1}: \sigma>1$ at level $\alpha=0.1$. It is given that $\chi^{2}_{15},_{0}._{1}=22.307, \chi^{2}_{15},_{0}._{9}=8.547, \chi^{2}_{30},_{0}._{1}=40.256, \chi^{2}_{30},_{0}._{9}=20.599$, where $P(W>\chi^{2}_{n},_{a})=\alpha$ and W~$\chi^{2}_{n}$. Then which of the following statements are true?

Let $X_{1},X_{2},\dots,X_{7}$ be a random sample drawn from a continuous distribution with unknown unique median M. The null hypothesis $H_{0}:M=2$ is tested against the alternative $H_{1}:M>2$ at level of significance 0.05 using the right-tailed test based on the Sign test statistic K, which is the number of observations in the sample greater than 2. If the observed sample is −3,−6,1,9,4,10,12, which of the following statements are true?

Let X be a random variable with the probability density function $f(x) = 2\theta{}x + 2(1 - \theta)(1 - x)$ if 0 < x < 1, and 0 otherwise, where $\theta \in [0, 1]$. Based on single observation x, the critical region of the most powerful test for testing null hypothesis $H_{0} : \theta = 1/2$ against alternative hypothesis $H_{1} : \theta = 1$, at level of significance $\alpha = 0.25$, is

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from a gamma distribution with shape parameter $\alpha > 0$ and scale parameter $\beta = 1$. For a suitable constant C, the rejection region of the most powerful test for testing $H_{0} : \alpha = 1$ against $H_{1} : \alpha = 2$ is of the form

Suppose $\varphi$ is a most powerful test of size 0.05 for testing a simple null hypothesis $H_{0}$ against a simple alternative hypothesis $H_{1}$. If the power of the test is 0.4, then which of the following is true?

Let $X_{1}, X_{2}, \dots, X_{5}$ be a random sample of size 5 from an absolutely continuous distribution having median M. Let S denote the number of $X_{i}'s$ greater than 0. For testing $H_{0} : M = 0$ against $H_{1} : M > 0$, let $\varphi(X) = 1$ if $S > c, \nu$ if S = c, and 0 if S < c be a test of size $\alpha = 0.05$, where $\nu \in [0, 1]$ and c ∈ {−1, 0, …, 5} are fixed constants. Then $c + \nu$ equals

Let $X_{1}, X_{2}$ be a random sample from a population having probability density function f ∈ {$f_{0}, f_{1}$} where $f_{0}(x) = 1/2$ if 0 ≤ x ≤ 2 and 0 otherwise, and $f_{1}(x) = 1/4$ if 0 ≤ x ≤ 4 and 0 otherwise. For testing the null hypothesis $H_{0} : f = f_{0}$ against the alternate hypothesis $H_{1} : f = f_{1}$, the power of a most powerful test of size $\alpha = 0.05$ is equal to

The life (in hours) of an electrical component is exponentially distributed with mean $\theta > 0$. For testing $H_{0} : \theta = 5/(\ln(10) - \ln(9))$ against $H_{1} : \theta = 5/\ln(2)$, three such components are chosen at random. Suppose that we reject the null hypothesis $H_{0}$ if and only if two or more of these three components survive for less than five hours. Then, which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{n}, Y_{n})$ be a random sample of size n from a bivariate distribution F(X,Y) with absolutely continuous marginal distribution functions F_X and F_Y of X and Y, respectively. Let r_S be the Spearman's rank correlation coefficient defined with ranks of $X_{i}$ and ranks of $Y_{i}, i = 1, 2, \dots, n$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_m$ and $Y_{1}, Y_{2}, \dots, Y_{n}$ be two mutually independent random samples from populations with absolutely continuous distribution functions F_X and F_Y, respectively. For N = m + n, define $T_N = \sum(i=1..N)$ iZ$_{i}$ where $Z_{i} = 1$ if the i-th observation in the combined ordered arrangement of N observations is from F_X; and $Z_{i} = 0$, otherwise. Then, which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from Uniform$[0, \theta]$ distribution, $\theta > 0$. Consider the likelihood ratio test of size 0.001 for testing $H_{0} : \theta = 3$ against $H_{1} : \theta \ne 3$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For i = 1, 2, 3, let $R_{i}$ denote the rank of |$X_{i}$| among |$X_{1}$|, |$X_{2}$| and |$X_{3}$|. If T⁺ $= \sum(i : X_{i}>0) R_{i}$ is the Wilcoxon signed-rank statistic, then 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?

Let $X_{1}, \dots, X_{7}$ and $Y_{1}, \dots, Y_{9}$ be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of $H_{0}: F = G$ vs $H_{1}: F \ne G$, let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

Let $X_{1}, \dots, X_{6}$ be a random sample from the gamma density f(x|$\lambda) = (\lambda^{4}/6)$ $e^{-\lambda{}x}$ $x^{3}$ for $x > 0, \lambda > 0$ unknown. Let $T = \sum{}X_{i}$ and $\psi$ be the UMP test of size 0.05 for $H_{0}: \lambda = 1$ against $H_{1}: \lambda > 1$. With $\chi^{2}_{\nu,\alpha}$ the $(1 - \alpha)-$th quantile of $\chi^{2}_\nu$, the test $\psi$ rejects $H_{0}$ iff

Under H: X ~ p with p(x) = 1/20, and under K: X ~ q with q(x) = x/210, x ∈ {1, …, 20}. Define test functions $\varphi(x) = 1$ if x ≤ 2 (else 0) and $\psi(x) = 1$ if x ≥ 19 (else 0). Which of the following statements are true?

Let $X_{1}, \dots, X_{25}$ be i.i.d. Bernoulli(p), X̄ their mean, $T_{1} = 5({\bar{X}} - 0.5)/\sqrt{{\bar{X}}(1 - {\bar{X}})}$ for 0 < X̄ < 1 (±5 at the endpoints) and $T_{2} = 10({\bar{X}} - 0.5)$. For $H_{0}: p = 0.5$ vs $H_{1}: p > 0.5$, test $\psi_{i}$ rejects iff $T_{i} > 2$. If the observed X̄ ∈ (0.5, 0.75), which of the following statements are true?

Let $X_{1}, \dots, X_{n}$ be a random sample from $N(\mu, 1)$. To test $H_{0}: \mu = \mu_{0}$ against $H_{1}: \mu > \mu_{0}$, consider (A) reject iff ${\bar{X}}_{n} > c_{1}$ with $P_{\mu_{0}}(X_{n} > c_{1}) = \alpha$, and (B) reject iff Median{$X_{1}, \dots, X_{n}$} $> c_{2}$ with $P_{\mu_{0}}(Median$ $> c_{2}) = \alpha$. Which of the following statements are true?

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be from $N(\mu, \sigma^{2})$ with both unknown, X̄ the mean and $S^{2}$ the sample variance. With $z_\alpha$ and $t_{m,\alpha}$ the $(1 - \alpha)-$th quantiles, which of the following represent 90% confidence intervals for $\mu$?

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0}: F = F_{0}$ vs $H_{1}: F \ne F_{0}$, consider $T_{1,n}$ = sup_x |$F_{n}(x) - F_{0}(x)$| and $T_{2,n}$ = n·sup_x |$F_{n}(x) - F_{0}(x)$|, where $F_{n}$ is the empirical CDF. Which of the following statements are true?