Answer: To determine the sample size needed to estimate the true proportion of homeowners who have vegetable gardens with a desired level of confidence, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the sample size needed
- Z is the Z-score corresponding to the desired level of confidence (typically 1.96 for a 95% confidence level)
- p is the estimated proportion of homeowners who have vegetable gardens (0.50 based on the given information)
- E is the desired margin of error (6 percentage points or 0.06)
Plugging in the values:
n = (1.96^2 * 0.50 * (1-0.50)) / 0.06^2
n = (3.8416 * 0.25) / 0.0036
n = 0.9604 / 0.0036
n ≈ 267.2778
Therefore, a sample size of approximately 267 is needed to estimate the true proportion of homeowners who have vegetable gardens to within 6 percentage points with a 95% confidence level.
None of the given answer choices (205, 281, 141, 83) match the calculated sample size.
Step-by-step explanation: