The standard error is higher when n = 125 compared to when n=500 because a sample statistic based on less data (i.e., smaller n) tends to be less accurate and have larger standard errors.

Question

The standard error is higher when n = 125 compared to when n=500 because a sample statistic based on less data (i.e., smaller n) tends to be less accurate and have larger standard errors.

BlackTom AI · Accepted Answer

Question restatement: The standard error is higher when n = 125 compared to when n = 500 because a sample statistic based on less data (i.e., smaller n) tends to be less accurate and have larger standard errors.

Option 1: True. The standard error of a statistic typically decreases as the sample size n increases, because standard error is often proportional to 1/√n. With smaller samples (n = 125), there is more sampling variability, which leads to larger standard errors and less precision. This aligns with fundamental sampling theory: more data generally yields more precise estimates.

Option 2: False. This would claim that smaller samples produce smaller standard errors, or that accuracy does not depend on n. In reality, decreasing n increases sampling variability, not decreases it, so the standard error tends to be larger for smaller samples. This contradicts the standard relationship between sample size and standard error.

Additional nuance: The statement emphasizes that a statistic based on less data tends to be less accurate. That is a correct intuition, since standard error is a measure of precision, and with fewer observations there is more variability in the estimate. Therefore, the claim that the standard error is higher for n = 125 than for n = 500 is consistent with the mathematics of standard error definitions across common statistics (e.g., sample mean, proportion) where SE ∝ 1/√n (up to constants depending on the distribution).

Potential misunderstandings to watch for: If one incorrectly assumes that standard error is a property of the statistic alone and not of the sample size, they might think SE is unaffected by n. That would be a misconception. Similarly, confusing standard error with standard deviation of the population can lead to errors; SE concerns the variability of the estimator across repeated samples, which decreases with larger n.

The standard error is higher when n = 125 compared to when n=500 because a sample statistic based on less data (i.e., smaller n) tends to be less accurate and have larger standard errors.判断题

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