We can use the stratified random sampling formula to calculate the overall sample mean and the population estimate for the mean salary ( $\bar{y}_{st}$ ). Here’s the process step by step:

1. Calculate the sample mean:

The sample mean ($\bar{y}$) is the weighted average of the means of each stratum (department), weighted by their sample sizes. The formula is:

$\bar{y} = \frac{1}{n} \sum_{i=1}^{L} n_i \bar{y}_i$

Where:

• $n$ = Total sample size = 150
• $n_i$ = Sample size of department $i$
• $\bar{y}_i$ = Sample mean of department $i$
• $L$ = Number of departments = 3

So, we plug in the values:

$\bar{y} = \frac{1}{150} \left( 60 \times 4000 + 55 \times 3500 + 35 \times 3000 \right)$

2. Calculate $\bar{y}_{st}$:

The population estimate for the mean salary ($\bar{y}_{st}$) is the weighted average of the sample means, weighted by the total number of employees in each stratum. The formula is:

$\bar{y}_{st} = \frac{1}{N} \sum_{i=1}^{L} N_i \bar{y}_i$

Where:

• $N$ = Total population = 1500
• $N_i$ = Number of employees in department $i$

Thus:

$\bar{y}_{st} = \frac{1}{1500} \left( 600 \times 4000 + 550 \times 3500 + 350 \times 3000 \right)$

Now let's calculate both $\bar{y}$ and $\bar{y}_{st}$.The calculations yield the following results:

• The sample mean ($\bar{y}$) is approximately 3583.33.
• The population estimate for the mean salary ($\bar{y}_{st}$) is also approximately 3583.33.

These values are the same because the sample sizes were proportionally allocated to each department.

Would you like further details or explanations on this?

Here are five related questions to consider:

1. How would the calculation change if the sample sizes were not proportional to the department sizes?
2. What are the advantages of using stratified random sampling in this scenario?
3. How would you calculate the variance of the sample mean?
4. How would the results change if department B had a significantly higher sample mean?
5. Can you generalize this process to other sampling techniques, such as cluster sampling?

Tip: Stratified sampling ensures that each subgroup (department) is adequately represented, improving the accuracy of the overall estimate.