(a) Find the most powerful test of size α(= 0·05) for testing H₀: μ = 0 vs. H₁: μ = 1, given a random sample of size 25 from N(μ, 16) population. (20 marks)

(b) A lot consists of some defective items. A random sample of 25 items has 6 defective item

Question

(a) Find the most powerful test of size α(= 0·05) for testing H₀: μ = 0 vs. H₁: μ = 1, given a random sample of size 25 from N(μ, 16) population. (20 marks)

(b) A lot consists of some defective items. A random sample of 25 items has 6 defective items with probability p₁ = θ and 19 non-defective items with probability p₂ = 1 – θ. Then estimate θ using the following:

(i) MLE method

(ii) Minimum χ²-method

(iii) Modified minimum χ²-method (15 marks)

(c) Differentiate between Mann-Whitney U-test and Wilcoxon sign test. The following data pertain to APGAR scores of 15 pregnant women in two care programmes A and B:

Programme A : 8 7 6 2 5 8 7 3

Programme B : 9 9 7 8 10 9 6

Is there a significant difference in APGAR scores of pregnant women under the two care programmes?

[Given, U₍₀.₀₅₎ = 10] (15 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) given its 20 marks, 30% to part (b) covering three estimation methods, and 30% to part (c) involving both differentiation and non-parametric testing. Structure as: (a) derive Neyman-Pearson lemma application with critical region, (b) present three estimation approaches with clear derivations, (c) tabulate differences then perform Mann-Whitney U-test with ranking and decision.
- Part (a): Apply Neyman-Pearson lemma to derive most powerful test; identify critical region as sample mean > k; compute critical value k = 0.4 + 1.645×(4/5) = 1.716 using Z-test; state rejection rule and power function
- Part (b)(i): Derive MLE as θ̂ = 6/25 = 0.24 using binomial likelihood L(θ) = θ⁶(1-θ)¹⁹ and log-likelihood differentiation
- Part (b)(ii): Set up minimum χ² by minimizing Σ(Oi-Ei)²/Ei; show equivalence to MLE in this case or derive modified form with expected frequencies 25θ and 25(1-θ)
- Part (b)(iii): Apply modified minimum χ² using weights in denominator; demonstrate Neyman modification or minimum logit χ² approach if applicable
- Part (c): Differentiate Mann-Whitney (two independent samples, ordinal/rank data) vs Wilcoxon signed-rank (paired/matched samples); correctly identify independent samples scenario here
- Part (c) computation: Rank pooled data (1-15), compute sum of ranks for smaller sample (Programme B, n=7), calculate U = 56 - 28 = 28 or U' = 49 - 28 = 21; compare with critical value U₀.₀₅ = 10; conclude no significant difference since U > 10

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all distributions and parameters: N(μ,16) with σ=4 known for (a), Binomial(25,θ) for (b), independent samples for Mann-Whitney in (c); states hypotheses with correct inequalities and significance level α=0.05	Identifies most distributions correctly but confuses paired vs independent samples in (c) or states hypotheses without proper notation; minor errors in parameter specification	Wrong distribution assumptions (e.g., uses t-test instead of Z-test in (a), treats (b) as Poisson); fails to state hypotheses or uses incorrect significance level
Method choice	20%	10	Applies Neyman-Pearson lemma correctly for (a); selects appropriate estimation methods for (b) with clear distinction between three approaches; correctly chooses Mann-Whitney U-test over Wilcoxon signed-rank for (c) with proper justification	Uses correct general methods but applies Neyman-Pearson without full justification; estimates θ using MLE correctly but confuses minimum χ² variants; chooses correct test in (c) without clear rationale	Uses likelihood ratio test instead of Neyman-Pearson for (a); applies method of moments or wrong estimation technique for (b); uses Wilcoxon signed-rank incorrectly for independent samples in (c)
Computation accuracy	20%	10	Accurate calculations throughout: critical value k=1.716 or Z=1.645 for (a); θ̂=0.24 with correct second-order condition for MLE, proper χ² minimization for (ii)-(iii); correct ranking, rank sums (RA=46, RB=59), U=17.5 or U=17, U'=17.5 for (c)	Minor arithmetic errors in critical value or rank sums; correct MLE but algebraic errors in χ² derivatives; ranking correct but U calculation formula slightly wrong	Major computational errors: wrong critical region, incorrect θ estimate, confused ranking with ties handled incorrectly, wrong U formula application leading to invalid conclusion
Interpretation	20%	10	Clear interpretation: states rejection region and power for (a); explains why MLE and minimum χ² coincide for (b); interprets U statistic correctly, compares with critical value, states p-value direction, and gives contextual conclusion about APGAR programmes	States results without full interpretation; mentions significance but lacks context; partial explanation of method equivalence in (b); conclusion present but not well-linked to maternal health context	No interpretation of results; fails to compare test statistic with critical value; no conclusion about hypothesis or practical significance; missing context entirely
Final answer & units	20%	10	All final answers boxed/highlighted: critical region X̄ > 1.716 with power 0.638; θ̂=0.24 (MLE), identical or properly derived modified estimate; U=17.5 (or 17), decision 'Do not reject H₀' with clear statement that no significant difference exists between programmes at 5% level	Most answers present but not clearly demarcated; missing power calculation or final decision statement; units correct but presentation inconsistent	Missing final answers or incorrect decisions; wrong conclusion about significance; no clear statement of test outcome; units omitted where relevant (e.g., APGAR score interpretation)

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