Agricultural Statistics

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 ($0 \lt P \le 1$).

Analysis of sample size dynamics:

• Incorrect: Non-sampling error may increase with large scale surveys, but sampling error decreases.
• Incorrect: Standard error $\text{SE} = \sigma / \sqrt{n}$ decreases as $n$ increases.
• Correct: Sampling error is inversely proportional to the square root of the sample size ($n$). As sample size increases, the sample better approximates the population parameters, thereby steadily decreasing sampling error.
• Incorrect: Sample size has a direct mathematical impact on sampling error.

Properties of relative measures of skewness:

• Standard relative measures of skewness (such as Karl Pearson's coefficient $S_k = \frac{\text{Mean} - \text{Mode}}{\sigma}$, Bowley's coefficient, and moment-based measure $\beta_1 = \frac{\mu_3^2}{\mu_2^3}$ or $\gamma_1 = \sqrt{\beta_1}$) are pure dimensionless numbers, completely independent of both change of origin and change of scale.
• Absolute measures (like $\mu_3$) depend on scale, but standard measures of skewness refer to relative measures which are invariant to scale.
• They are invariant to origin.
• They are independent of both origin and scale.