CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $P(W_{2}$ˣ $< W_{3}$ʸ) = 11/16

2. B

   $P(W_{1}$ˣ $< W_{1}$ʸ) = 1/2

3. C

   $P(W_{2}$ˣ $< W_{3}$ʸ) = 13/16

4. D

   $P(W_{1}$ˣ $< W_{1}$ʸ) = 1/4

Multi-select · full marks only for the exact set · Estimation

1. A

   X̄ is the method of moments estimator of $\theta$

2. B

   $X_{(1)}$ is the maximum likelihood estimator of $\theta$

3. C

   $X_{(1)} - 1/n$ is the uniformly minimum variance unbiased estimator of $\theta$

4. D

   $X_{(1)}$ is a sufficient statistic for $\theta$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   If $\psi_{1}$ rejects $H_{0}$, then $\psi_{2}$ also rejects $H_{0}$

2. B

   If $\psi_{1}$ does not reject $H_{0}$, then $\psi_{2}$ also does not reject $H_{0}$

3. C

   If $\psi_{2}$ rejects $H_{0}$, then $\psi_{1}$ also rejects $H_{0}$

4. D

   If $\psi_{2}$ does not reject $H_{0}$, then $\psi_{1}$ also does not reject $H_{0}$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   The test in (A) is the uniformly most powerful test of size $\alpha$

2. B

   The test in (B) is the uniformly most powerful test of size $\alpha$

3. C

   $P_\mu({\bar{X}}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $\mu > \mu_{0}$

4. D

   $P_{\mu_{0}}(Median${$X_{1}, \dots, X_{n}$} $> \mu_{0}) = 1/2$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $({\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.05}$, ${\bar{X}} + (S/\sqrt{n})$ $t_{n-1,0.05})$

2. B

   $({\bar{X}} - (\sigma/\sqrt{n})$ $z_{0.05}$, ${\bar{X}} + (\sigma/\sqrt{n})$ $z_{0.05})$

3. C

   $[{\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9}$, $\infty)$

4. D

   $(-\infty, {\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9})$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $T_{1,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$

2. B

   $T_{2,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$

3. C

   lim P_F($T_{2,n}$ > 1) = 1 for all F

4. D

   $T_{2,n}$ converges in distribution to a degenerate random variable under $H_{0}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $(1/\sum{}n_{i}) \sum_{i}\sum_{j} Y_{ij}$ is an unbiased estimator of $\mu$

2. B

   $2\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function

3. C

   $\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function

4. D

   $(1/n_{2}) \sum_{j} (Y_{2j} -$ Ȳ$_{2})$ is an unbiased estimator of $\alpha_{2}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $(1/\sigma^{2}) Y^{T}X\beta$̂ has a central $\chi^{2}_{p+1}$ distribution

2. B

   $(1/\sigma^{2})(Y - X\beta$̂$)^{T}(Y - X\beta$̂) has a central $\chi^{2}_{n-p-1}$ distribution

3. C

   $X\beta$̂ and $(Y - X\beta$̂$)^{T}(Y - X\beta$̂) are independently distributed

4. D

   $(1/\sigma^{2}) \sum(Y_{i} -$ Ȳ$)^{2}$ has a central $\chi^{2}_{n-1}$ distribution

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   $a_{22}$ ~ $\chi^{2}_{3}$

2. B

   ½ $a_{22}$ ~ $\chi^{2}_{5}$

3. C

   $(1/33)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{3}$

4. D

   $(1/9)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{5}$

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 0$

2. B

   $S_\rho = (6/\pi) \sin$⁻$^{1}(\rho/2)$

3. C

   If $S_\rho = 0$, then $X_{1}$ and $Y_{1}$ are independent

4. D

   If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 1/2$

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   E(M) = 4

2. B

   $E(M^{2}) = 17$

3. C

   $E(M^{3}) = 72$

4. D

   Var(M) = 1

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   X̄ = 95

2. B

   $S^{2} = 77$

3. C

   C > 180/19

4. D

   R > 8