Estimated standard error is used when...

Estimated standard error is used when...

• A. The data are nonparametric.
• B. The sample size is extremely large.
• C. The sample mean is biased.
• D. The true population standard deviation is unknown.

Answer: (D)

The standard error represents how far the sample mean tends to be from the true population mean across repeated samples. When the population standard deviation is known, you can compute the exact standard error as σ/√n. In practice, σ is almost always unknown, so we estimate the standard error by using the sample standard deviation: s/√n. This estimated standard error is what underlies t-based inferences, since replacing σ with s makes the sampling distribution heavier-tailed and leads to the t distribution with n−1 degrees of freedom. So the reason the estimated standard error is used is that the true population standard deviation is unknown. If the population SD were known, you wouldn’t need to estimate it. The other ideas don’t fit: nonparametric contexts involve different assumptions and typically different ways to assess variability; extremely large samples don’t remove the need to estimate σ when it’s unknown; bias of the sample mean concerns the location estimate, not the spread captured by the standard error.

The standard error represents how far the sample mean tends to be from the true population mean across repeated samples. When the population standard deviation is known, you can compute the exact standard error as σ/√n. In practice, σ is almost always unknown, so we estimate the standard error by using the sample standard deviation: s/√n. This estimated standard error is what underlies t-based inferences, since replacing σ with s makes the sampling distribution heavier-tailed and leads to the t distribution with n−1 degrees of freedom.

So the reason the estimated standard error is used is that the true population standard deviation is unknown. If the population SD were known, you wouldn’t need to estimate it. The other ideas don’t fit: nonparametric contexts involve different assumptions and typically different ways to assess variability; extremely large samples don’t remove the need to estimate σ when it’s unknown; bias of the sample mean concerns the location estimate, not the spread captured by the standard error.