Optimal Tests and Likelihood Ratios - Testing of Hypotheses - Statistics Previous Year Questions

1

2019 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2019

A \(2^3\) factorial experiment with factors A, B and C is arranged in two blocks of four plots each as follows: (Below (1) denotes the treatment in which A, B and C are at the lower level, ac denotes the treatment in which A and C are at the higher level and B is at the lower level and so on.)

Block 1	(1)	ab	ac	bc
Block 2	a	b	c	abc

The treatment contrast that is confounded with the blocks is

A

BC

B

AC

C

AB

D

ABC

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2

2019 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2019

Let \( X \) be a random variable with probability density function \( f(x; \theta) = \theta e^{-\theta x} \), where \( x \geq 0 \) and \( \theta > 0 \). To test \( H_0: \theta = 1 \) against \( H_1: \theta > 1 \), the following test is used: Reject \( H_0 \) if and only if \( X > \log_e 2 \). Then the size of the test is ...

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3

2019 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2019

Let \(X_1\) be a random sample of size 1 from uniform distribution over \((\theta, \theta^2)\), where \(\theta > 1\). To test \(H_0: \theta = 2\) against \(H_1: \theta = 3\), reject \(H_0\) if and only if \(X_1 > 3.5\). Let \(\alpha\) and \(\beta\) be the size and the power, respectively, of this test. Then \(\alpha + \beta\) (rounded off to two decimal places) is equal to ...

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4

2020 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2020

Let \(X_1, \ldots, X_n\) be a random sample of size \(n \ (\geq 2)\) from an exponential distribution with the probability density function \(f(x; \theta) = \begin{cases} \frac{1}{\theta} e^{-x/\theta}, & x > 0 \\ 0, & \text{otherwise} \end{cases}\) where \(\theta \in \{1, 2\}\). Consider the problem of testing \(H_0: \theta = 1\) against \(H_1: \theta = 2\), based on \(X_1, \ldots, X_n\). Which of the following statements is TRUE?

A

Likelihood ratio test at level \(\alpha \ (0 < \alpha < 1)\) leads to the same critical region as the corresponding most powerful test at the same level.

B

(Only (A) is fully visible; other options are cut off)

C

D

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5

2020 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2020

Let $X_1, \ldots, X_n$ be a random sample of size $n$ ($n \geq 2$) from $N(\theta, 1)$ distribution, where $\theta \in (-\infty, \infty)$. Consider the problem of testing $H_0: \theta \in [1, 2]$ against $H_1: \theta < 1$ or $\theta > 2$, based on $X_1, \ldots, X_n$. Which of the following statements is TRUE? Critical region, of level $\alpha$ $(0 < \alpha < 1)$ of uniformly most powerful test for $H_0$ against $H_1$ is of the form $\{(x_1, \ldots, x_n): c_1 \leq \sum_{i=1}^n x_i \leq c_2\}$, where $c_1$ and $c_2$ are such that the test is of level $\alpha$

A

Critical region, of level $\alpha$ $(0 < \alpha < 1)$ of uniformly most powerful test for $H_0$ against $H_1$ is of the form $\{(x_1, \ldots, x_n): \sum_{i=1}^n x_i > c$ or $\sum_{i=1}^n x_i < d\}$, where $c$ and $d$ are such that the test is of level $\alpha$

B

At any level $\alpha \in (0, 1)$, uniformly most powerful test for $H_0$ against $H_1$ does NOT exist

C

At any level $\alpha \in (0, 1)$, the power of uniformly most powerful test for $H_0$ against $H_1$ is less than $\alpha$

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6

2020 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2020

Let \(X\) be a discrete random variable with probability mass function \(f \in \{f_0, f_1\}\), where

	\(x = 1\)	\(x = 2\)	\(x = 3\)	\(x = 4\)	\(x = 5\)
\(f_0(x)\)	0.10	0.10	0.10	0.10	0.60
\(f_1(x)\)	0.05	0.06	0.08	0.09	0.72

The power of the most powerful level \(\alpha = 0.1\) test for testing \(H_0: X \sim f_0\) against \(H_1: X \sim f_1\), based on \(X\), equals ____________________ (correct up to two decimal places).

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7

2020 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2020

Let \(X_1, ..., X_5\) be a random sample from a distribution with the probability density function \[ f(x; \theta) = \frac{1}{2} e^{-|x - \theta|}, x \in (-\infty, \infty), \] where \(\theta \in (-\infty, \infty)\). For testing \(H_0: \theta = 0\) against \(H_1: \theta < 0\), let \(\sum_{i=1}^5 Y_i\) be the sign test statistic, where \[ Y_i = \begin{cases} 1, & X_i > 0 \\ 0, & \text{otherwise} \end{cases}. \] Then the size of the test, which rejects \(H_0\) if and only if \(\sum_{i=1}^5 Y_i \le 2\), equals ____________________ (correct up to one decimal place).

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8

2021 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2021

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n\) (≥ 2) from a \(N(0, \sigma^2)\) distribution. For a given \(\sigma > 0\), let \(f_\sigma\) denote the joint probability density function of \((X_1, X_2, \ldots, X_n)\) and \(S = \{f_\sigma : \sigma > 0\}\). Let \(T_1 = \sum_{i=1}^n X_i^2\) and \(T_2 = \left(\frac{1}{n} \sum_{i=1}^n X_i\right)^2\). For any positive integer \(\nu\) and any \(\alpha \in (0, 1)\), let \(\chi_{\nu, \alpha}^2\) denote the \((1 - \alpha)\)-th quantile of the central chi-square distribution with \(\nu\) degrees of freedom. Consider testing \(H_0 : \sigma = 1\) against \(H_1 : \sigma > 1\) at level \(\alpha\). Then which one of the following statements is true?

A

\(S\) has a monotone likelihood ratio in \(T_1\) and \(H_0\) is rejected if \(T_1 > \chi_{n, \alpha}^2\)

B

\(S\) has a monotone likelihood ratio in \(T_1\) and \(H_0\) is rejected if \(T_1 > \chi_{n, 1-\alpha}^2\)

C

\(S\) has a monotone likelihood ratio in \(T_2\) and \(H_0\) is rejected if \(T_2 > \chi_{n, \alpha}^2\)

D

\(S\) has a monotone likelihood ratio in \(T_2\) and \(H_0\) is rejected if \(T_2 > \chi_{n, 1-\alpha}^2\)

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9

2021 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2021

Let \(\{0,2\}\) be a realization of a random sample of size 2 from a binomial distribution with parameters 2 and \(p\), where \(p \in (0,1)\). To test \(H_0: p = \frac{1}{2}\) against \(H_1: p = \frac{1}{2}\), the observed value of the likelihood ratio test statistic equals __________ (round off to 2 decimal places).

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10

2021 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2021

Let \( X \) be a discrete random variable with probability mass function \( p \in \{p_0, p_1\} \), where

\( x \)	7	8	9	10
\( p_1(x) \)	0.69	0.10	0.16	0.05
\( p_0(x) \)	0.90	0.05	0.04	0.01

To test \( H_0: p = p_0 \) against \( H_1: p = p_1 \), the power of the most powerful test of size 0.05, based on \( X \), equals ________ (round off to 2 decimal places).

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11

2022 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2022

A random sample $X_1, X_2, \ldots, X_6$ of size 6 is taken from a Bernoulli distribution with the parameter $\theta$. The null hypothesis $H_0: \theta = \frac{1}{2}$ is to be tested against the alternative hypothesis $H_1: \theta > \frac{1}{2}$, based on the statistic $Y = \sum_{i=1}^6 X_i$. If the value of $Y$ corresponding to the observed sample values is 4, then the p-value of the test statistic is

A

$\frac{21}{32}$

B

$\frac{9}{64}$

C

$\frac{11}{32}$

D

$\frac{7}{64}$

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12

2022 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2022

Let \(X_1, X_2, X_3, X_4\) be a random sample of size four from a Bernoulli distribution with the parameter \(\theta\), \(0 < \theta < 1\). Consider the null hypothesis \(H_0: \theta = \frac{1}{4}\) against the alternative hypothesis \(H_1: \theta > \frac{1}{4}\). Suppose \(H_0\) is rejected if and only if \(X_1 + X_2 + X_3 + X_4 > 2\). If \(\alpha\) is the probability of Type I error for the test and \(\gamma(\theta)\) is the power function of the test, then the value of \(16\alpha + 7 \gamma\left(\frac{1}{2}\right)\) (in integer) is equal to ________

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13

2022 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2022

Given that \(\Phi(1.645) = 0.95\) and \(\Phi(2.33) = 0.99\), where \(\Phi(\cdot)\) denotes the cumulative distribution function of a standard normal random variable. For a random sample \(X_1, X_2, \dots, X_n\) from a normal population \(N(\mu, 2^2)\), where \(\mu\) is unknown, the null hypothesis \(H_0: \mu = 10\) is to be tested against the alternative hypothesis \(H_1: \mu = 12\). Suppose that a test that rejects \(H_0\) if the sample mean \(\bar{X}\) is large, is used. Then the smallest value of \(n\) (in integer) such that Type I error is 0.05 and Type II error is at most 0.01, is ________

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14

2023 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2023

Suppose that \( x \) is an observed sample of size 1 from a population with probability density function \( f(\cdot) \). Based on \( x \), consider testing
\( H_0: f(y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} ; y \in \mathbb{R} \) against \( H_1: f(y) = \frac{1}{2} e^{-|y|} ; y \in \mathbb{R} \).
Then which one of the following statements is true?

A

The most powerful test rejects \(H_0 \) if \(|x| > c \) for some \(c > 0 \)

B

The most powerful test rejects \(H_0 \) if \(|x| < c \) for some \(c > 0 \)

C

The most powerful test rejects \(H_0 \) if \(||x| - 1| > c \) for some \(c > 0 \)

D

The most powerful test rejects \(H_0 \) if \(||x| - 1| < c \) for some \(c > 0 \)

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15

2023 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2023

Suppose that \((X, Y)\) has joint probability mass function
\(P(X = 0, Y = 0) = P(X = 1, Y = 1) = \theta,\)
\(P(X = 1, Y = 0) = P(X = 0, Y = 1) = \frac{1}{2} - \theta,\)
where \(0 \le \theta \le \frac{1}{2}\) is an unknown parameter. Consider testing \(H_0: \theta = \frac{1}{4}\) against \(H_1: \theta = \frac{1}{2}\), based on a random sample \((X_1, Y_1), (X_2, Y_2), \ldots, (X_n, Y_n)\) from the above probability mass function. Let \(M\) be the cardinality of the set \(\{i : X_i = Y_i, 1 \le i \le n\}\). If \(m\) is the observed value of \(M\), then which one of the following statements is true?

A

The likelihood ratio test rejects \(H_0\) if \(m > c\) for some \(c\)

B

The likelihood ratio test rejects \(H_0\) if \(m < c\) for some \(c\)

C

The likelihood ratio test rejects \(H_0\) if \(c_1 < m < c_2\) for some \(c_1\) and \(c_2\)

D

The likelihood ratio test rejects \(H_0\) if \(m < c_1\) or \(m > c_2\) for some \(c_1\) and \(c_2\)

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16

2023 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2023

Let \(X\) be a random sample of size \(1\) from a population with cumulative distribution function \[F(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases}\] where \(\theta > 0\) is an unknown parameter. To test \(H_0: \theta = 1\) against \(H_1: \theta = 2\), consider using the critical region \(\{x \in \mathbb{R} : x < 0.5\}\). If \(\alpha\) and \(\beta\) denote the level and power of the test, respectively, then \(\alpha + \beta\) (rounded off to two decimal places) equals ______________

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17

2024 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2024

In a testing of hypothesis problem, which one of the following statements is true?

A

The probability of the Type-I error cannot be higher than the probability of the Type-II error

B

Type-II error occurs if the test accepts the null hypothesis when the null hypothesis is actually false

C

Type-I error occurs if the test rejects the null hypothesis when the null hypothesis is actually false

D

The sum of the probability of the Type-I error and the probability of the Type-II error should be 1

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18

2024 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2024

A random sample of size 40 is drawn from a population having four distinct categories as \(i = 1, 2, 3, 4\). The data are given as

Category	1	2	3	4
Observed Frequency	5	8	12	15

Let \(\theta_i\) be the probability that an observation comes from the \(i\)-th category, \(i = 1, 2, 3, 4\). If the chi-square goodness-of-fit test is used to test \(H_0: \theta_i = \frac{1}{4}\), \(i = 1, 2, 3, 4\) against \(H_1: \theta_i \neq \frac{1}{4}\) for some \(i = 1, 2, 3, 4\), then which one of the following statements is true?

A

Under \(H_0\), the test statistic follows central chi-square distribution with 3 degrees of freedom and the observed value of the test statistic is 5.8

B

Under \(H_0\), the test statistic follows central chi-square distribution with 3 degrees of freedom and the observed value of the test statistic is 1.4

C

Under \(H_0\), the test statistic follows central chi-square distribution with 4 degrees of freedom and the observed value of the test statistic is 5.8

D

Under \(H_0\), the test statistic follows central chi-square distribution with 4 degrees of freedom and the observed value of the test statistic is 1.4

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19

2024 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2024

Let \(X\) be a random sample of size one from a population having \(N(0, \sigma^2)\) distribution, where \(\sigma > 0\) is an unknown parameter. Let \(\Phi(\cdot)\) denote the cumulative distribution function of a standard normal random variable and let \(\chi^2_{\nu, \alpha}\) denote the \((1 - \alpha)\)-th quantile of the central chi-square distribution with \(\nu\) degrees of freedom. It is given that \(\Phi(1.96) = 0.975\), \(\Phi(1.64) = 0.95\), \(\chi^2_{1, 0.05} = 3.841\), \(\chi^2_{2, 0.05} = 5.991\). To test \(H_0: \sigma^2 = 1\) against \(H_1: \sigma^2 = 2\), using the Neyman-Pearson most powerful test of size \(0.05\), the critical region is given by \(\lambda(X) > c\), where \(c \geq 0\) is a constant and \(\lambda(X) = \frac{f(x; \sigma^2 = 2)}{f(x; \sigma^2 = 1)},\) where \(f(x; \sigma^2)\) is the probability density function of a \(N(0, \sigma^2)\) distribution. Then the value of \(c\) equals __________ (rounded off to two decimal places).

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20

2024 · Statistics · Testing of Hypotheses · Optimal Tests and Likelihood Ratios

Statistics (ST) 2024

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n (\ge 2)\) from a population having probability density function \[ f(x; \lambda) = \begin{cases} \frac{1}{\lambda} e^{-\frac{x}{\lambda}} & \text{if } x \ge 0 \\ 0 & \text{otherwise}, \end{cases} \] where \(\lambda > 0\) is an unknown parameter. Let \(T_1 = \sum_{i=1}^n X_i\) and \(T_2 = (\sum_{i=1}^n X_i)^{-1}\). For any positive integer \(\nu\) and any \(\alpha \in (0,1)\), let \(\chi^2_{\nu;\alpha}\) denote the \((1-\alpha)\)-th quantile of the central chi-square distribution with \(\nu\) degrees of freedom. Consider testing \(H_0: \lambda = \lambda_0\) against \(H_1: \lambda > \lambda_0\). Then which of the following tests is/are uniformly most powerful test at level \(\alpha\)?

A

\(H_0\) is rejected if \(\frac{2}{\lambda_0} T_1 > \chi^2_{2n;\alpha}\)

B

\(H_0\) is rejected if \(\frac{2}{\lambda_0} T_1 > \chi^2_{2n;1-\alpha}\)

C

\(H_0\) is rejected if \(\frac{\lambda_0}{2} T_2 > \chi^2_{2n;\alpha}\)

D

\(H_0\) is rejected if \(\frac{\lambda_0}{2} T_2 > \chi^2_{2n;1-\alpha}\)

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