The statement "In simple random sampling without replacement, the sample mean square is an unbiased estimate of the population mean square" is incorrect. Let me clarify the reasoning.

Explanation:

• Population Mean Square (PMS): This refers to the variance of the population, denoted by $\sigma^2$.

• Sample Mean Square (SMS): This is the variance calculated from a sample, and it is typically denoted by $s^2$.

Key points:

1. Bias in Sample Variance: When we calculate the sample variance using $s^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$, this formula tends to underestimate the true population variance $\sigma^2$. Therefore, it is a biased estimator of $\sigma^2$.

2. Unbiased Estimate: To correct this bias, the formula for the unbiased sample variance is modified to: $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$ This correction ($n-1$ instead of $n$ in the denominator) accounts for the fact that we're estimating the variance using a sample and not the full population.

3. Simple Random Sampling Without Replacement: When sampling is done without replacement, the sample variance calculated as $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$ becomes an unbiased estimator of the population variance $\sigma^2$.

Conclusion:

The sample mean square, as defined without the correction (i.e., dividing by $n$), is not an unbiased estimator of the population mean square. However, the corrected sample variance formula ($s^2$ with $n-1$) is an unbiased estimate of $\sigma^2$ when using simple random sampling without replacement.

Would you like further clarification or examples?

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Questions to Expand Understanding:

1. What is the difference between sampling with and without replacement in terms of variance estimation?
2. How is the correction factor $(n-1)$ derived for the unbiased sample variance?
3. Can you explain how the finite population correction factor affects variance estimation?
4. What is the difference between the population mean and sample mean, and how does this affect variance calculations?
5. How do unbiased estimators differ from consistent estimators in statistics?

Tip:

Always distinguish between sample statistics (used to estimate) and population parameters (true values) in statistical analysis.