If the events A₁, A₂, , An are independent and P(Ai) = (1/1+i)... | Agrigyan

• Probability with Combinations:
• Composition of eligible officers:
• Production department: 3
• Purchase department: 4
• Sales department: 2
• Chartered Accountant (CA): 1
• Total pool size: $N = 3 + 4 + 2 + 1 = 10$ individuals
• Committee size: $k = 4$ individuals
• Direct Symmetry Method:
• Since each of the 10 individuals has an equal chance of being chosen in a committee of 4, the probability that any specific individual (the CA) is included is simply:

• Combinatorial Verification:
• Total number of possible committees: $\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$
• Favorable committees (including the 1 CA, remaining 3 chosen from the other 9): $\binom{1}{1} \times \binom{9}{3} = 1 \times \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$
• Probability: $P = \frac{84}{210} = \frac{2}{5}$
• Conclusion: The probability is 2/5 (Option 4).

Correct option: (4) 215

• Variance Stabilizing Transformations:
• When analyzing proportional data ($p$) where observations follow a binomial distribution, the variance is proportional to $p(1 - p)$, which depends heavily on the mean.
• Angular / Arcsine Transformation:
• The variance-stabilizing transformation for binomial proportions or percentages is the sine inverse transformation ($f(p) = \arcsin(\sqrt{p})$ or $\sin^{-1}\sqrt{p}$).
• This stabilizes the variance to a constant value of approximately $\frac{1}{4n}$ (in radians), satisfying the homoscedasticity assumption of ANOVA.
• Other Transformations:
• Square root transformation ($\sqrt{x}$ or $\sqrt{x + 0.5}$): Used for count data following a Poisson distribution.
• Logarithmic transformation ($\log x$): Used when standard deviation is proportional to the mean.
• Fisher's $z$-transformation: Used to normalize the sample correlation coefficient $r$.
• Conclusion: The appropriate transformation is Sine inverse transformation (Option 2).

Correct option: (2) Sine inverse transformation

• Neyman-Pearson Fundamental Lemma:
• In hypothesis testing theory, the Neyman-Pearson Lemma provides the necessary and sufficient condition for constructing the Most Powerful (MP) test of size $\alpha$ for testing a simple null hypothesis against a simple alternative hypothesis:

• The optimal rejection region is defined by the likelihood ratio test statistic:

• When the family of distributions possesses a Monotone Likelihood Ratio (MLR), this lemma generalizes to construct Uniformly Most Powerful (UMP) tests for one-sided composite hypotheses.
• Conclusion: The Neyman-Pearson lemma constructs the Most powerful test (Option 2).

Correct option: (2) Most powerful test