一家公司调查员工对公司政策的了解程度。

该公司有 6000 名全职员工和 4000 名兼职员工。

(a) 说明如何抽取 200 名员工的分层样本。

(b) 说明与简单随机样本相比，使用分层样本的一个优点。

从 80 名全职员工和一个独立的 80 名兼职员工随机样本中，给他们做了一项政策认知测试。结果如下表所示。

(c) 清楚写出假设，在 1% 的显著性水平下检验全职和兼职员工的平均政策认知分数是否不同。

(d) 说明中央极限定理对 (c) 中检验的意义。

(e) 写出在 (c) 中进行检验时所作出的一个假设。

在所有员工完成培训课程后，这 80 名全职员工和 80 名兼职员工又进行了另一项政策认知测试。检验统计量 z 的值为 2.53。

(f) 在 1% 的显著性水平下，根据该结果评论全职和兼职员工的政策认知情况。

(g) 解释 (c) 和 (f) 的答案。

按全职与兼职两层在总体中的比例分配 200 个名额，分别得到 120 与 80。随后必须在每一层内部独立编号并用随机数抽取，不能只写出人数而省略随机抽样步骤。

The required numbers from the two strata are

Label the full-time staff from 1 to 6000 and the part-time staff from 1 to 4000. Use random numbers to select 120 different labels from the full-time list and 80 different labels from the part-time list.

分层抽样按总体结构分配样本，保证全职与兼职员工都按其人口比例得到代表，避免简单随机样本偶然对某一类抽得过多或过少。

A stratified sample reflects the proportions of full-time and part-time staff in the workforce, so both types of staff are appropriately represented. This can reduce sampling variability compared with a simple random sample.

两个样本独立且样本量都为 80，可对两个总体均值之差作双尾大样本检验。用两组样本方差估计总体方差，先计算差值的标准误，再把 $z$ 统计量与 $1\%$ 双尾临界值 $\pm2.5758$ 比较。

Let $\mu_f$ and $\mu_p$ be the population mean awareness scores for full-time and part-time staff respectively.

The estimated standard error of the difference between the sample means is

Therefore, the test statistic is

For a two-tailed test at the $1\%$ significance level, the critical values are $\pm2.5758$. Since

the result is significant and $H_0$ is rejected. There is evidence, at the $1\%$ significance level, of a difference between the mean policy awareness scores of full-time and part-time staff.

Equivalently, the two-tailed p-value is approximately $0.00468$, which is less than $0.01$ and gives the same conclusion.

中央极限定理关注的是样本均值的抽样分布，而不是要求每名员工的原始分数本身服从正态分布。两组样本量均为 80，因此两组样本均值都可近似看作正态分布，从而支持 (c) 的正态检验。

Since both sample sizes are large, the Central Limit Theorem implies that the sampling distributions of the mean score for full-time staff and the mean score for part-time staff are both approximately normal, even if the underlying score distributions are not normal. Hence their independent difference is also approximately normal, justifying the normal test used in part (c).

计算标准误时把样本方差 $21$、$19$ 分别当作两总体方差的估计，因此所作假设是两组样本方差可代替对应的总体方差。

For the purpose of this test, it is assumed that each sample variance is equal to the corresponding population variance; that is,

培训后仍使用 (c) 的 $1\%$ 双尾检验标准。给定的 $z=2.53$ 没有超过正临界值 $2.5758$，所以结果不显著，不能拒绝两类员工平均认知分数相同的原假设。

Using the same two-tailed $1\%$ test as in part (c), the critical values are $\pm2.5758$. Since

the result is not significant and $H_0$ is not rejected. There is insufficient evidence, at the $1\%$ significance level, of a difference between the mean policy awareness scores of full-time and part-time staff after training.

(c) 表明培训前两类员工的平均政策认知分数有显著差异，而 (f) 表明培训后已没有显著差异；结合两次检验，可解释为培训缩小并消除了原先可检测到的差距。

Before the training course there was a significant difference between the mean awareness scores of full-time and part-time staff, but after training there was no significant difference. Therefore, the training course appears to have closed the gap between the two types of staff.