Probability, Statistics & Linear Programming

Hypothesis Testing

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1. Neyman–Pearson lemma and UMP tests

Read this as a chapter →

Lec-22 Neyman Pearson Fundamental Lemma

NPTEL · Statistical Inference

Lec-24 UMP Tests

NPTEL · Statistical Inference

Monotone likelihood ratio ⇒ one-sided UMP; two-sided usually has none.

Neyman–Pearson and UMP tests free

In plain words

Fix how often you wrongly reject a true null, then buy as much power as that allows. For a two-sided alternative there is usually no single best test at all.

Neyman–Pearson lemma

For $H_{0}: \theta = \theta_{0}$ vs $H_{1}: \theta = \theta_{1} ($ both simple), the most powerful size $-\alpha$ test rejects when

$L(\theta_{1})/L(\theta_{0}) > k$ , with k chosen so the size is $\alpha$ .

From MP to UMP

If the family has monotone likelihood ratio in T (exponential families do), then the one-sided test "reject if T > c" is UMP for $H_{0}: \theta \le \theta_{0}$ vs $H_{1}: \theta > \theta_{0}$ .

Two-sided alternatives generally admit no UMP test — the best one-sided tests point in opposite directions. One then uses UMP unbiased tests or the likelihood ratio test.

Direction matters

If larger $\theta$ makes T smaller, the rejection region flips. For a Gamma $(4, \lambda)$ sample, $T = \sum{}X_{i}$ has $2\lambda{}T$ ~ $\chi^{2}_{4n}$ , and larger $\lambda$ shrinks T, so the UMP test for $\lambda > 1$ rejects for small T.

Size and power

Size = P(reject | $H_{0})$ ; power = P(reject | $H_{1}) = 1 - \beta$ .
Power increases with n, with | $\theta_{1} - \theta_{0}$ |, and with $\alpha$ .
A test with power < size is worse than random (biased test).

Likelihood ratio test

$\Lambda =$ $sup_{\Theta_{0}}$ $L / sup_\Theta L$ ; reject for small $\Lambda$ . Under regularity, $-2 \ln \Lambda \to \chi^{2}_r$ where r = the number of restricted parameters (Wilks). This is the general-purpose test when no UMP exists.

Key takeaways

NP: simple vs simple, likelihood ratio, threshold from the size.
MLR ⇒ one-sided UMP; two-sided usually has none.
Check whether the statistic increases or decreases with the parameter before writing the rejection region.

Type I against Type II: the trade-off you cannot escapeinteractivefree

Exam focus: NP gives the most powerful test for simple-vs-simple; monotone likelihood ratio upgrades it to UMP for one-sided alternatives. Two-sided alternatives usually have no UMP test.

Practice

Part B2025 Juneshow ▾

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}

A. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} > (1/2)\chi^{2}_{2n,\alpha}$ + n ln 2
B. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} < (1/2)\chi^{2}_{2n,1-\alpha}$ + n ln 2✓ Correct
C. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} > \chi^{2}_{n,\alpha}$ + n ln 2
D. $\sum_{i}$ ₌ $_{1}^{n} \ln X_{i} < \chi^{2}_{n,1-\alpha}$ + n ln 2

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

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?

A. $H_{0}$ is rejected
B.The p-value of the test is greater than 0.01✓ Correct
C.Under $H_{0}, E(K) = 3.5$ ✓ Correct
D.If $X_{(i)}$ denotes the i-th smallest observation of the sample, then $[X_{(2)},X_{(6)}]$ is a confidence interval for M with confidence coefficient at least 0.95

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

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

A.x < 1/4
B.x > 3/4✓ Correct
C.1/2 < x < 3/4
D.1/4 < x < 1/2

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

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?

A. $(1 - \varphi)$ is a most powerful test at level 0.6 for testing $H_{1}$ against $H_{0}$ .✓ Correct
B. $(1 - \varphi)$ is a most powerful test at level 0.4 for testing $H_{1}$ against $H_{0}$ .
C. $(1 - \varphi)$ is a most powerful test at level 0.05 for testing $H_{1}$ against $H_{0}$ .
D. $(1 - \varphi)$ is NOT a most powerful test for testing $H_{1}$ against $H_{0}$ at any level.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

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

A. $\prod(i=1..n) X_{i} > C$ ✓ Correct
B. $\sum(i=1..n) X_{i} > C$
C. $\prod(i=1..n) X_{i} < C$
D. $\sum(i=1..n) X_{i} < C$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

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

A.0.4625
B.0.5425
C.0.7625✓ Correct
D.0.6225

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

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?

A.The size of the test is 0.01.
B.The power of the test is 0.5.✓ Correct
C.The test is unbiased at level $\alpha = 0.05$ .✓ Correct
D.If all of these three components survive for less than five hours, then the p-value of the test is 0.001.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾

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?

A.Size of the test $\varphi$ is 0.1.✓ Correct
B.Size of the test $\psi$ is 0.05.
C.(Power of the test $\psi) > 0.05$ .✓ Correct
D.(Power of the test $\psi) > ($ Power of the test $\varphi)$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

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

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?

A.If $\psi_{1}$ rejects $H_{0}$ , then $\psi_{2}$ also rejects $H_{0}$
B.If $\psi_{1}$ does not reject $H_{0}$ , then $\psi_{2}$ also does not reject $H_{0}$ ✓ Correct
C.If $\psi_{2}$ rejects $H_{0}$ , then $\psi_{1}$ also rejects $H_{0}$ ✓ Correct
D.If $\psi_{2}$ does not reject $H_{0}$ , then $\psi_{1}$ also does not reject $H_{0}$

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

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?

A.The test in (A) is the uniformly most powerful test of size $\alpha$ ✓ Correct
B.The test in (B) is the uniformly most powerful test of size $\alpha$
C. $P_\mu({\bar{X}}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $\mu > \mu_{0}$ ✓ Correct
D. $P_{\mu_{0}}(Median$ { $X_{1}, \dots, X_{n}$ } $> \mu_{0}) = 1/2$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

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

A.T ≥ ½ $\chi^{2}_{48, 0.05}$
B.T ≤ ½ $\chi^{2}_{48, 0.95}$ ✓ Correct
C.T ≥ ½ $\chi^{2}_{24, 0.05}$
D.T ≤ ½ $\chi^{2}_{24, 0.95}$

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

For testing

H_{0} : \theta = \theta_{0}

against

H_{1} : \theta = \theta_{1}

at level

\alpha

, which of the following are true?

A.The Neyman–Pearson likelihood-ratio test is most powerful among all tests of its level.✓ Correct
B.Enlarging $\alpha$ cannot decrease the power of the best level $-\alpha$ test.✓ Correct
C.The p-value is the probability that $H_{0}$ is true given the observed data.
D.In a one-parameter family with monotone likelihood ratio, the one-sided test is UMP against the one-sided composite alternative.✓ Correct

Solution

(1) is the Neyman–Pearson lemma.

(2) A larger $\alpha$ admits more rejection regions, and the optimum over a larger class is no worse.

(3) is the classic misreading — the p-value is computed assuming $H_{0}$ and is not a posterior probability of $H_{0}$ .

(4) is the Karlin–Rubin theorem.

Save for spaced revision (free account)

Part Bshow ▾

In hypothesis testing, a Type I error is

A.rejecting $H_{0}$ when $H_{0}$ is true✓ Correct
B.accepting $H_{0}$ when $H_{0}$ is false
C.rejecting $H_{1}$ when $H_{1}$ is true
D.any error whose probability exceeds 0.05

The worked solution and the per-option analysis are in Trap Analysis.

2. Likelihood ratio and standard tests

Read this as a chapter →

Lec-31 Likelihood Ratio Tests - I

NPTEL · Statistical Inference

Lec-36 Test for Goodness of Fit

NPTEL · Statistical Inference

Lec-39 Confidence Intervals

NPTEL · Statistical Inference

Watch the quantile convention — that is where marks are lost.

Standard tests and confidence intervals free

In plain words

Compare the best fit the null allows with the best fit available overall. Twice the negative logarithm behaves like a chi-square, with degrees of freedom counting the restrictions imposed.

Which test

Situation	Statistic	Distribution
One mean, $\sigma$ known	$({\bar{X}} - \mu)/(\sigma/\sqrt{n})$	N(0,1)
One mean, $\sigma$ unknown	$({\bar{X}} - \mu)/(S/\sqrt{n})$	$t_{n-1}$
Two means, equal variances	pooled t	$t_{n_{1}+n_{2}-2}$
Paired data	t on differences	$t_{n-1}$
One variance	$(n-1)S^{2}/\sigma^{2}$	$\chi^{2}_{n-1}$
Two variances	$S_{1}^{2}/S_{2}^{2}$	$F_{n_{1}-1, n_{2}-1}$
Goodness of fit	$\sum(O-E)^{2}/E$	$\chi^{2}_{k-1-p}$
Independence in an r × c table	same	$\chi^{2}_{(r-1)(c-1)}$
ANOVA	MS_between/MS_within	$F_{k-1, N-k}$

Confidence intervals

Two-sided $100(1-\alpha)$ %: X̄ ± $t_{n-1, \alpha/2}\cdot{}S/\sqrt{n}$ .

The quantile trap. If $t_{m,\alpha}$ denotes the $(1-\alpha)-$ th quantile, then $t_{n-1,0.9}$ is the 10th percentile (negative). So $- ({\bar{X}} - (S/\sqrt{n})t_{n-1,0.05}$ , ${\bar{X}} + (S/\sqrt{n})t_{n-1,0.05})$ is a 90% two-sided interval; $- (-\infty, {\bar{X}} - (S/\sqrt{n})t_{n-1,0.9})$ is a 90% one-sided interval; $- [{\bar{X}} - (S/\sqrt{n})t_{n-1,0.9}$ , $\infty)$ is only a 10% interval.

Read the quantile convention in the question before choosing.

Nonparametric tests

Test	Use
Sign test	median
Wilcoxon signed rank	symmetric location
Mann–Whitney U	two samples
Wald–Wolfowitz runs	two samples, any difference
Kolmogorov–Smirnov	sup\| $F_{n} - F_{0}$ \|, scaled by $\sqrt{n}$

Runs distribution (m X's, n Y's): P(R = 2k) = 2C(m−1,k−1)C(n−1,k−1)/C(m+n,m); the odd case adds the two mixed terms.

Glivenko–Cantelli: sup| $F_{n} - F$ | → 0 a.s. So $\sqrt{n}\cdot{}\sup$ converges (Kolmogorov limit), and $n\cdot{}\sup \to \infty$ .

Key takeaways

$- \sigma$ known $\Rightarrow z; \sigma$ unknown ⇒ t with n − 1 degrees of freedom.

One-sided intervals use the other tail — check the quantile convention.
KS scales as $\sqrt{n}$ ; multiplying by n diverges.

What 95% confidence actually meansinteractive

This visual is part of Trap Analysis

Forty samples, forty intervals, and a count of how many cover the truth — the misconception this kills is worth several marks.

Unlock interactive visuals

Exam focus: Match the situation to the test and remember the degrees of freedom. For confidence intervals, watch whether the quantile is upper or lower and whether σ is known.

Practice

Part B2025 Decembershow ▾

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}

X_{1}+X_{2} > 4/5

. Then, the probability of type-I error is

A.8/25
B.13/25
C.17/25✓ Correct
D.22/25

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

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?

A.The maximum possible value of T is 115
B.Right-tailed test based on T is appropriate for testing $H_{0}$ against $H_{1}$
C.Under $H_{0}, E(T) = 76$
D.Under $H_{0}, P(R_{1} = 1, R_{9} = 16) = 1/240$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

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?

A.If the observed value of X̄ is 0.6, then $\varphi$ does not reject $H_{0}$ ✓ Correct
B.If the observed value of X̄ is 1.6, then $\varphi$ rejects $H_{0}$ ✓ Correct
C.If the observed value of X̄ is 1.3, then $\varphi$ rejects $H_{0}$
D.If the observed value of X̄ is 1.2, then $\varphi$ does not reject $H_{0}$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

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?

A.The likelihood function is maximized at $\theta = 1/(x - 1)$ ✓ Correct
B.The maximum value of the likelihood function is $e^{-1}/(x$ − 1)✓ Correct
C.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if (x − $1)e^{-x}$ < k, for some k > 0✓ Correct
D.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if x > c, for some c > 1

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

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?

A.p-value of the sign test is 0.04
B. $H_{0}$ is NOT rejected at 5% level of significance by the sign test✓ Correct
C.The observed value of Wilcoxon signed rank test statistic W⁺ is equal to 10✓ Correct
D.If $P_(H_{0})(W$ ⁺ ≥ 14) = 0.06, then $H_{0}$ is rejected at 5% level of significance by the Wilcoxon signed rank test

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

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?

A.If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then E(r_S) = 0, ∀n ≥ 2.✓ Correct
B.If n = 3, then P(r_S = 0) = 0.✓ Correct
C.If F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then Var(r_S) = 1/n, ∀n ≥ 2.
D.If n = 4 and F(X,Y)(x, y) = F_X(x)F_Y(y), ∀(x, y), then P(r_S = 0) = 1/24.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

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

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

A.11/25
B.49/6
C.103/25✓ Correct
D.53/6

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

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?

A. $H_{0}$ is rejected at 5% level of significance✓ Correct
B. $H_{0}$ is rejected at 1% level of significance
C. $H_{0}$ is not rejected at 5% level of significance
D.Observed value of the test statistic is 12.5✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

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?

A.If max{ $X_{1}, X_{2}, X_{3}$ } is 3.1, then $H_{0}$ is rejected.✓ Correct
B.If max{ $X_{1}, X_{2}, X_{3}$ } is 1.3, then $H_{0}$ is rejected.
C.If max{ $X_{1}, X_{2}, X_{3}$ } is 0.1, then $H_{0}$ is rejected.✓ Correct
D.The power of the test at $\theta = 0.3$ is 1.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

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?

A.P(T⁺ = 3) = 1/4✓ Correct
B.Var(T⁺) = 7/2✓ Correct
C.P(T⁺ > 3) = 5/8
D.P(T⁺ > 4) = 1/8

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

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)

_{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?

A.If F_X(x) = F_Y(x) ∀x, then E(T_N) = m(N + 1)/2.✓ Correct
B.If F_X(x) = F_Y(x) ∀x, then Var(T_N) = mn(N + 1)/24.
C.If F_X(x) = F_Y(x) ∀x, then the distribution of T_N is symmetric about mn/2.
D.The minimum and maximum possible values of T_N are m(m + 1)/2 and N(N + 1)/2 − m(m + 1)/2, respectively.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

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}

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?

A. $T_{1,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$ ✓ Correct
B. $T_{2,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
C.lim P_F( $T_{2,n}$ > 1) = 1 for all F✓ Correct
D. $T_{2,n}$ converges in distribution to a degenerate random variable under $H_{0}$

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

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

H_{1}: F \ne G

, let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

A. $P_{H_{0}}(R$ = 6) = 28/286, $P_{H_{0}}(R$ = 9) = 28/143.
B. $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 15/286.
C. $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 28/143.✓ Correct
D. $P_{H_{0}}(R$ = 6) = 21/286, $P_{H_{0}}(R$ = 9) = 15/286.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

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

A. $({\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.05}$ , ${\bar{X}} + (S/\sqrt{n})$ $t_{n-1,0.05})$ ✓ Correct
B. $({\bar{X}} - (\sigma/\sqrt{n})$ $z_{0.05}$ , ${\bar{X}} + (\sigma/\sqrt{n})$ $z_{0.05})$
C. $[{\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9}$ , $\infty)$
D. $(-\infty, {\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9})$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let

X_{1}, \dots, X_{n}

i.i.d. N(\mu, \sigma^{2})

, both parameters unknown, with sample mean X̄ and sample variance

S^{2}

. Which of the following are true?

A. $\sqrt{n}({\bar{X}} - \mu)/S$ has a t-distribution with n − 1 degrees of freedom.✓ Correct
B. $(n - 1)S^{2}/\sigma^{2}$ has a chi-squared distribution with n − 1 degrees of freedom.✓ Correct
C.X̄ and $S^{2}$ are independent.✓ Correct
D.All three of the above hold for i.i.d. samples from any distribution with finite variance.

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

A 95% confidence interval for a normal mean with known

\sigma

has width proportional to

\sigma/\sqrt{n}

. To halve the width, the sample size must be

A.doubled
B.quadrupled✓ Correct
C.halved
D.increased by 50%

Solution

Width ∝ $1/\sqrt{n}$ , so halving it needs $\sqrt{n}$ to double, i.e. n to quadruple — the square-root law that makes extra precision expensive.

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