位置1的问题 From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately 题库 1/1[input]. (Note: your answer should retain THREE decimal places.)简答题
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Question

位置1的问题 From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately 题库 1/1[input]. (Note: your answer should retain THREE decimal places.)简答题

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BlackTom AI · Accepted Answer

We are given a population of size N = 500 with population variance σ^2 = 900, so the standard deviation is σ = sqrt(900) = 30. A sample of size n = 225 is drawn. The standard error of the mean depends on whether we apply the finite population correction (FPC).

First, without FPC (simple random sampling from an infinite population):
- Standard error (SE) = σ / sqrt(n) = 30 / sqrt(225) = 30 / 15 = 2.000.

Second, with finite population correction (appropriate for sampling without replacement from a finite population):
- FPC factor = sqrt((N - n) / (N - 1)) = sqrt((500 - 225) / (500 - 1)) = sqrt(275 / 499) ≈ sqrt(0.551102) ≈ 0.7424.
- Adjusted SE = (σ / sqrt(n)) * FPC = 2.000 * 0.7424 ≈ 1.4848.
- When rounded to three decimal places, this becomes 1.485.

A note on the given numeric result: if someone reports approximately 1.535, that would correspond to a slightly larger correction factor than the standard FPC here or a different interpretation of the sampling scheme. Using the conventional finite population correction for the given N and n yields about 1.485 to three decimals.

Summary of the key computations:
- σ = sqrt(900) = 30
- Without FPC: SE = 30 / sqrt(225) = 2.000
- With FPC: SE = 2.000 * sqrt((500 - 225) / (500 - 1)) ≈ 2.000 * 0.7424 ≈ 1.485

Since the answer options are not provided in the input, these steps show how to compute the standard error under both common interpretations and what the three-decimal result would be under the finite population correction.