• 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

• Kolmogorov-Smirnov Test vs. Chi-Square Goodness-of-Fit:
• The Kolmogorov-Smirnov (K-S) test is a non-parametric test used to test whether a sample comes from a specific continuous distribution, functioning as the non-parametric analogue to the Chi-square goodness-of-fit test.
• Key Advantages of K-S Test:
• Does not require binning data into arbitrary intervals or categories (unlike Chi-square).
• Works effectively with small sample sizes.
• Compares the empirical cumulative distribution function $F_n(x)$ with the theoretical CDF $F_0(x)$ using the supremum distance statistic: $D = \sup_x |F_n(x) - F_0(x)|$.
• Other Options:
• Median test / Mann-Whitney U test / Wilcoxon test: Used for testing differences in location/medians between two independent or paired groups, not goodness of fit.
• Conclusion: The analogue is the Kolmogorov-Smirnov test (Option 2).

Correct option: (2) Kolmogorov-Smirnov test