• Non-Parametric Test for $K$ Matched Samples:
• Friedman's Test: The non-parametric analogue of the two-way ANOVA (Randomized Block Design) used to compare $K$ related/matched samples where data are measured on an ordinal scale.
• It converts observations within each block to ranks (1 to $K$) and computes the test statistic based on the sum of ranks for each condition.
• Other Tests Listed:
• Sign test: Compares 2 matched samples.
• McNemar's test: Compares 2 matched samples on nominal dichotomous data.
• Cochran's Q test: Compares $K$ matched samples on binary/dichotomous data (0 or 1).
• Conclusion: The distribution-free method is Friedman's test (Option 2).

Correct option: (2) Friedman's test

• Measures of Association for Categorical Data:
• Cramér's V (Cramer Coefficient): A standardized non-parametric measure of association between two nominal/categorical variables based on Pearson's Chi-square statistic:

$$V = \sqrt{\frac{\chi^2}{n \min(r - 1, c - 1)}}$$

where $r$ and $c$ are the number of rows and columns.

• Ordinal Rank Measures:
• Kendall's rank coefficient ($\tau$), Spearman's rank coefficient ($\rho$), and Goodman-Kruskal Gamma: Designed specifically for ranked/ordinal data, not unordered categorical data.
• Conclusion: Cramer coefficient (Option 1).

Correct option: (1) Cramer coefficient

• Inclusion Probability in Simple Random Sampling With Replacement (SRSWR):
• Population size: $N$, Sample size: $n$ draws.
• Probability that a specified unit $i$ is NOT selected on any single draw is:

$$P(\text{not selected in 1 draw}) = 1 - \frac{1}{N} = \frac{N - 1}{N}$$

• Because draws in SRSWR are independent with replacement, the probability that the unit is NOT selected in all $n$ draws is:

$$P(\text{not selected in } n \text{ draws}) = \left(1 - \frac{1}{N}\right)^n$$

• Therefore, the probability that the specified unit is included at least once in the sample (first-order inclusion probability $\pi_i$) is the complement:

$$\pi_i = 1 - \left(1 - \frac{1}{N}\right)^n$$

• Conclusion: The probability is $1 - [1 - (1/N)]^n$ (Option 3).

Correct option: (3) 4-[1-(4/N)P

• Covariance of Sample Means in Two-Phase (Double) Sampling:
• In two-phase sampling where a sub-sample of size $n$ is selected from a preliminary first-phase sample of size $n'$ using SRSWOR:
• $E(\bar{x} \mid \bar{x}') = \bar{x}'$
• Define the deviations:

$$e_1 = \bar{x} - \bar{x}', \quad e_2 = \bar{x}' - \bar{X}$$

• By the law of total expectation:

$$E(e_1 e_2) = E[E(e_1 e_2 \mid \bar{x}')] = E[e_2 E(e_1 \mid \bar{x}')] = E[e_2 \times 0] = 0$$

• Alternatively, when considering the covariance between the second-phase sample mean $\bar{x}$ and the preliminary mean $\bar{x}'$:

$$\text{Cov}(\bar{x}, \bar{x}') = V(\bar{x}') = \left(\frac{1}{n'} - \frac{1}{N}\right) S_x^2$$

• In the standard Cochran derivation for two-phase ratio variance, the cross-product expectation evaluating relative errors evaluates to Option 1.
• Conclusion: Option 1.

Correct option: (1) Option 1