Answer:
To determine the sample size (\(n\)) needed for the confidence interval, you can use the formula:
\[ n = \left( \frac{Z^2 \times p \times (1 - p)}{E^2} \right) \]
where:
- \( Z \) is the Z-score corresponding to the desired confidence level (for 95% confidence, \( Z \approx 1.96 \)),
- \( p \) is the estimated proportion (32.3% or 0.323 in decimal form),
- \( E \) is the margin of error (0.02).
Substitute these values into the formula:
\[ n = \left( \frac{(1.96)^2 \times 0.323 \times (1 - 0.323)}{(0.02)^2} \right) \]
Now, calculate this expression to find the required sample size.
Certainly! Let's proceed with the calculation:
\[ n = \frac{(1.96)^2 \times 0.323 \times (1 - 0.323)}{(0.02)^2} \]
\[ n = \frac{3.8416 \times 0.323 \times 0.677}{0.0004} \]
\[ n \approx \frac{0.82713}{0.0004} \]
\[ n \approx 2067.825 \]
So, the sample size needed for the 95% confidence interval to be no more than 0.02 is approximately 2068 males aged 25 or older in Peoria. Since you can't have a fraction of a person, you'd need to round up to the nearest whole number.
