The sample size needed to estimate a population mean to within 2 units with 95% confidence when the population standard deviation equals 8 is:当总体标准差等于 8 时，以 95% 的置信度将总体均值估计在 2 个单位以内所需的样本量为：

Question

BlackTom AI · Accepted Answer

To determine the sample size required for estimating a population mean within a specified margin of error, we use the formula n = (Z_{alpha/2} * sigma / E)^2.
First, identify the components: the confidence level is 95%, which gives Z_{alpha/2} ≈ 1.96 for a two-sided interval; the population standard deviation sigma is 8; and the desired margin of error E is 2.
Plugging these values in: n = (1.96 * 8 / 2)^2 = (1.96 * 4)^2 = (7.84)^2 = 61.4656.
Since sample size must be a whole number and we typically round up to ensure the margin of error is achieved, the required n is 62.
Now evaluate the options:
- Option a: 61. This is slightly less than the calculated 61.4656, which would not guarantee the margin of error of 2 at 95% confidence. The logic here would understate the needed sample size, risking a larger actual margin of error.
- Option b: 62. This matches the ceiling of the calculation 61.4656, and thus is the correct rounded-up sample size to meet the specification.
- Option c: 9. This is far too small given the formula and would yield a substantially larger margin of error than 2, failing the requirement.
- Option d: 8. Similar to option c, this is unrealistically small for any reasonable estimation precision at 95% confidence with sigma = 8.
In summary, the calculation yields 61.4656, and rounding up to satisfy the margin of error leads to 62. The smaller values do not meet the target precision, while the larger ones are technically unnecessary given the rounding rule and the computation.

The sample size needed to estimate a population mean to within 2 units with 95% confidence when the population standard deviation equals 8 is:当总体标准差等于 8 时，以 95% 的置信度将总体均值估计在 2 个单位以内所需的样本量为：Single choice

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