In the context of confidence intervals for a population variance ${\sigma }^2$, can a (population or sample) variance ever be negative?

• A. Yes; variance can be negative when the sample mean is negative.
• B. Yes; variance can be negative if the sample size is small (e.g., $n<30$).
• C. Yes; variance can be negative when the confidence level is above 95%.
• D. No; variance is always nonnegative, so it cannot be negative.

Answer: (D)

Verified step by step guidance

1. Recall the definition of variance: it measures the average squared deviation of data points from the mean. Since it involves squaring differences, these squared values are always zero or positive.
2. Understand that both population variance (\(\sigma^{2}\)) and sample variance (\(s^{2}\)) are calculated using squared terms, which means they cannot be negative by their mathematical construction.
3. Recognize that the sample mean being negative does not affect the variance's sign because variance depends on squared deviations, not the sign of the mean itself.
4. Note that the sample size (whether small or large) does not cause variance to be negative; it may affect the accuracy or reliability of the variance estimate but not its sign.
5. Understand that confidence levels (such as 95% or higher) relate to the interval estimation around the variance but do not imply that the variance itself can be negative.