Which of the following is not a property of the sampling distribution of the $\mathrm{variance}$?

• A. Its mean equals the population $\mathrm{variance}$.
• B. It is used to estimate the population $\mathrm{proportion}$.
• C. It is approximately normal for large sample sizes if the population is normal.
• D. Its shape depends on the sample size and the underlying population distribution.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand what a sampling distribution of the variance represents. It is the probability distribution of the sample variance calculated from all possible samples of a given size drawn from a population.
2. Step 2: Recall the key properties of the sampling distribution of the variance: (a) Its mean equals the population variance, (b) Its shape depends on the sample size and the population distribution, and (c) For a normal population, the distribution of the sample variance follows a chi-square distribution, which becomes approximately normal for large sample sizes.
3. Step 3: Analyze each given statement in the problem: (i) 'Its mean equals the population variance' is true by definition, (ii) 'It is approximately normal for large sample sizes if the population is normal' aligns with the central limit theorem applied to the variance, (iii) 'Its shape depends on the sample size and the underlying population distribution' is also true because the distribution changes with these factors.
4. Step 4: Identify the statement that does not describe a property of the sampling distribution of the variance. The statement 'It is used to estimate the population proportion' is incorrect because the sampling distribution of the variance relates to variance estimation, not proportion estimation.
5. Step 5: Conclude that the property 'It is used to estimate the population proportion' is not a property of the sampling distribution of the variance, as it pertains to a different parameter (proportion) and a different sampling distribution.