第二部分：统计 [60分] 某公司有 1500 名员工。35% 的员工处于 21-40 岁年龄段，50% 处于 41-60 岁年龄段，其余处于 60 岁及以上年龄段。 将抽取一个包含 50 名员工的样本来参与一项关于员工士气的调查。生成一份按姓名首字母顺序排列的员工名单。选择一个从 1 到 30 的随机数 $k$。名单上的第 $k$ 个员工被选中，此后，名单上每隔 30 个员工将被选中，直到总共选中 50 名员工。

1. State the disadvantage of the systematic sampling method in this context.

The context provides the percentages of employees in different age groups, implying that age is a relevant factor affecting staff morale. Since the list is ordered alphabetically, systematic sampling does not guarantee that the sample will have a proportional representation of these age groups. Therefore, the sample may not be representative of the population.

1. State the appropriate sampling method to ensure representative age group proportions.

The method is stratified sampling.

2. Determine the proportional sample sizes for each stratum.

Divide the employees into 3 strata based on their age groups. The sample sizes should be proportional to the population sizes:

• 21-40 age group: $0.35 \times 50 = 17.5 \approx 18$
• 41-60 age group: $0.50 \times 50 = 25$
• $\ge 60$ age group: $0.15 \times 50 = 7.5 \approx 7$
  (Note: 17 from the 21-40 group and 8 from the $\ge 60$ group is also acceptable, provided the total is 50).

3. Describe the sampling mechanism within each stratum.

For each age group, obtain a list of the respective employees. Assign a unique number to each employee and use a random number generator to select the required number of employees independently (i.e., perform a simple random sample within each stratum).