Statistical Methods & Experimental Designs

To test the null hypothesis $H_0: \rho = 0$ (that the population correlation coefficient is zero) against $H_1: \rho \ne 0$, the Student's t-test statistic is:

This statistic follows a Student's t-distribution with $(n - 2)$ degrees of freedom, because two parameters (the means of $X$ and $Y$) are estimated from the bivariate sample.

For dependent/paired bivariate samples $(X_i, Y_i)$ of size $n$, the differences $d_i = X_i - Y_i$ are computed. The test statistic is:

where $\bar{d} = \frac{1}{n}\sum d_i$ and $s_d^2 = \frac{1}{n-1}\sum (d_i - \bar{d})^2$.
This follows Student's t-distribution with $(n - 1)$ degrees of freedom. Hence, Option 2 is correct.

While the asymptotic (large-sample) mean of Pearson's chi-square test for an $r \times s$ contingency table equals its degrees of freedom $(r - 1)(s - 1)$, J.B.S. Haldane (1939) derived the exact mean under conditional multinomial distribution with fixed marginal totals as:

Notice that as sample size $N \to \infty$, $\frac{N}{N-1} \to 1$, recovering the standard asymptotic mean $(r - 1)(s - 1)$. Thus, Option 3 is correct.

Analysis of sampling criteria:

• Incorrect: If probability equals zero, the element cannot be sampled, violating probability sampling axioms.
• Incorrect: Population size $N$ is an integer count, not a valid probability ($P \le 1$).
• Incorrect: Sample size $n$ is an integer count, not a probability value.
• Correct: By definition of probability sampling, every individual item in the population has a known, non-zero probability of being included in the sample, which strictly lies between zero and one ($0 \lt P \le 1$).