Using a 5% margin of error, how many families should be included in the survey? Use Slovin’s formula: n = N / (1 + N * e^2). Using proportional allocation, how many from each group... Using a 5% margin of error, how many families should be included in the survey? Use Slovin’s formula: n = N / (1 + N * e^2). Using proportional allocation, how many from each group should be taken as samples? Read more

Steps to Solve

1. Determine Total Population (N)

Identify the total population size you are sampling from. For this example, let's assume the total population of families is ( N = 1000 ) families.

2. Identify Desired Margin of Error (E)

Select the desired margin of error for the survey. For example, if the desired margin of error is 5%, convert this to decimal form:

$$ E = 0.05 $$

3. Apply Slovin's Formula

Use Slovin's formula to calculate the required sample size (n):

$$ n = \frac{N}{1 + N \cdot E^2} $$

Substituting the values:

$$ n = \frac{1000}{1 + 1000 \cdot (0.05)^2} $$

4. Calculate Sample Size

Perform the calculations in the formula:

First, calculate ( N \cdot E^2 ):

$$ N \cdot E^2 = 1000 \cdot 0.0025 = 2.5 $$

Now use this to solve for n:

$$ n = \frac{1000}{1 + 2.5} = \frac{1000}{3.5} \approx 285.71 $$

Since you can’t have a fraction of a family, round to the nearest whole number:

$$ n \approx 286 $$

5. Proportional Allocation from Income Groups

If you have different income groups and their proportions, allocate the sample size proportionally. For example, if you have 3 income groups:

• Group A: 50% (500 families)
• Group B: 30% (300 families)
• Group C: 20% (200 families)

Calculate the sample size for each group:

• Sample from Group A: $$ n_A = 286 \cdot 0.5 \approx 143 $$

• Sample from Group B: $$ n_B = 286 \cdot 0.3 \approx 86 $$

• Sample from Group C: $$ n_C = 286 \cdot 0.2 \approx 57 $$

6. Final Allocation

The final allocation of families surveyed from each income group would be:

• Group A: 143 families
• Group B: 86 families
• Group C: 57 families