To determine the sample size required using the given parameters, we use the formula for a stratified sample:
1. Determine the total number of homes and the number in urban and rural areas:
Total homes: N = 50523
Urban homes: N_1 = 24882
Rural homes: N_2 = 50523 - 24882 = 25641
2. Proportion of the sample from each stratum (urban and rural):
Urban: W_1 = \frac{N_1}{N} = \frac{24882}{50523}
Rural: W_2 = \frac{N_2}{N} = \frac{25641}{50523}
3. Use the formula for the sample size with a given confidence level and margin of error, the formula for sample size n is:
n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right)
where:
- Z is the Z-value for a 95% confidence level (1.96),
- p is the estimated proportion (0.60),
- E is the desired margin of error (0.06).
4. Calculate the sample size:
n = \left( \frac{1.96^2 \cdot 0.60 \cdot (1-0.60)}{0.06^2} \right)
n = \left( \frac{1.96^2 \cdot 0.60 \cdot 0.40}{0.0036} \right)
n \approx 256
5. Allocate the sample size proportionally among strata (rounded to the nearest whole number):
Urban: n_1 = W_1 \cdot n = \frac{24882}{50523} \cdot 256 \approx 126
Rural: n_2 = W_2 \cdot n = \frac{25641}{50523} \cdot 256 \approx 130
Answer: Total sample size is 256 , with 126 homes from the urban area and 130 homes from the rural area.