Step-by-step explanation:
The formula to calculate the minimum sample size for estimating a population proportion is:
\[ n = \frac{{Z^2 \cdot p \cdot (1 - p)}}{{E^2}} \]
Where:
- \(n\) = sample size needed
- \(Z\) = Z-score corresponding to the desired confidence level
- \(p\) = estimated proportion (expressed as a decimal)
- \(E\) = margin of error (expressed as a decimal)
For a 99% confidence level, the Z-score is approximately 2.576 (from standard normal distribution tables).
Given:
- Margin of error (E) = 0.03 (3 percentage points expressed as a decimal)
- Estimated proportion (p) = 0.35 (35% expressed as a decimal)
Plugging in these values:
\[ n = \frac{{2.576^2 \cdot 0.35 \cdot (1 - 0.35)}}{{0.03^2}} \]
Calculating this yields:
\[ n = \frac{{6.635776 \cdot 0.35 \cdot 0.65}}{{0.0009}} \]
\[ n = \frac{{1.8647416}}{{0.0009}} \]
\[ n \approx 2071.935 \]
Rounded up to the nearest whole number, the minimum sample size required is approximately 2072.
None of the provided options match this calculated value. However, considering the closest value, 2579 from the given options, might be used in practical situations that require rounding up or adjustment to ensure a sufficient sample size.
