Understanding Proportional Allocation in Sampling

This question involves determining the sample sizes for different schools using the proportional allocation technique. Proportional allocation is a method used in stratified sampling where the sample size for each stratum (in this case, each school) is made proportional to the size of that stratum in the population. This ensures that the representation of each school in the sample mirrors its representation in the total student population.

Data Breakdown for Sampling

First, let's list the information provided:

• Total number of students in the population (N): 480
• Total sample size to be selected (n_total): 60
• Number of students in each school (stratum):
  • School A (N₁): 120 students
  • School B (N₂): 200 students
  • School C (N₃): 160 students

We need to find the sample size for each school: n₁, n₂, and n₃.

Calculating Sample Sizes Using Proportional Allocation

The formula for proportional allocation is:

\[ n_i = \frac{N_i}{N} \times n_{total} \]

Where:

• n_i is the sample size for school i
• N_i is the total number of students in school i
• N is the total number of students across all schools
• n_total is the total sample size

School A Sample Size (n₁) Calculation

For School A:

• N₁ = 120
• N = 480
• n_total = 60

Using the formula:

\[ n_1 = \frac{120}{480} \times 60 \]

First, simplify the fraction \(\frac{120}{480}\):

\[ \frac{120}{480} = \frac{12}{48} = \frac{1}{4} \]

Now, calculate n₁:

\[ n_1 = \frac{1}{4} \times 60 = 15 \]

So, the sample size for School A is 15.

School B Sample Size (n₂) Calculation

For School B:

• N₂ = 200
• N = 480
• n_total = 60

Using the formula:

\[ n_2 = \frac{200}{480} \times 60 \]

Simplify the fraction \(\frac{200}{480}\):

\[ \frac{200}{480} = \frac{20}{48} = \frac{5 \times 4}{12 \times 4} = \frac{5}{12} \]

Now, calculate n₂:

\[ n_2 = \frac{5}{12} \times 60 \]

\[ n_2 = 5 \times \frac{60}{12} = 5 \times 5 = 25 \]

So, the sample size for School B is 25.

School C Sample Size (n₃) Calculation

For School C:

• N₃ = 160
• N = 480
• n_total = 60

Using the formula:

\[ n_3 = \frac{160}{480} \times 60 \]

Simplify the fraction \(\frac{160}{480}\):

\[ \frac{160}{480} = \frac{16}{48} = \frac{1 \times 16}{3 \times 16} = \frac{1}{3} \]

Now, calculate n₃:

\[ n_3 = \frac{1}{3} \times 60 = 20 \]

So, the sample size for School C is 20.

Final Sample Allocation

The calculated sample sizes for the three schools are:

• School A (n₁): 15
• School B (n₂): 25
• School C (n₃): 20

The total sample size is \(15 + 25 + 20 = 60\), which matches the required total sample size.

Comparing these results with the given options, the correct sample sizes (n₁, n₂, n₃) are 15, 25, 20 respectively.

Therefore, the correct option is the one that lists the sample sizes as 15, 25, and 20 for schools A, B, and C, respectively.