Question

For estimating a population characteristic, why is an unbiased statistic with a small standard error preferred over an unbiased statistic with a larger standard error?

Short answer

An unbiased statistic with a small standard error is preferred over one with a larger standard error because it provides a more precise and reliable estimate of the population characteristic. A small standard error means our estimate is closer to the true population parameter.

Step-by-step solution

Understanding Key Concepts

Begin by defining the three key statistical concepts central to the question: unbiased statistic, standard error, and population characteristic. An unbiased statistic is an accurate statistic that does not overestimate or underestimate the population parameter. Standard error is a measure of the statistical accuracy of an estimate, or the standard deviation of the sampling distribution. A population characteristic (or parameter) is a measure that describes a population, such as a population mean or a population proportion.

Examining the Difference in Standard Errors

Now, we need to consider what the difference is between small and large standard errors for an unbiased statistic. The smaller the standard error, the closer the sample statistic is likely to be to the population parameter. Conversely, a larger standard error could lead to the sample statistic being farther from the population parameter.

Explaining the Preference

Finally, we need to explain why a small standard error is preferred in estimating a population characteristic. A smaller standard error indicates a more precise estimate of the population parameter. Since the goal is to estimate the population parameter as accurately as possible, an unbiased statistic with a small standard error would give a better, more reliable estimate than one with a large standard error as it means our sample mean is closer to the actual population mean.