Section B: Statistics [60 marks] A company has 1500 employees. 35% of employees are in the 21-40 age group, 50% are in the 41-60 age group, and the rest are in the 60 and above age group. A sample of 50 employees is to be chosen to take part in a survey on staff morale. A list of the employees in alphabetical order by name is generated. A random number $k$ from 1 to 30 is chosen. The $k$th employee on the list is chosen and thereafter, every $30^{\text{th}}$ employee on the list will be chosen until a total of 50 employees are chosen.

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).