Questions: You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion. You would like to be 99.5% confident that your estimate is within 2.5% of the true population proportion. How large of a sample size is required? n=3152 Do not round mid-calculation. However, use of a critical value rounded to three decimal places will suffice.

Solution Steps

For a confidence level of \( 99.5\% \), the critical value \( Z \) is approximately \( 2.807 \).

Since there is no reasonable estimate for the population proportion, we use \( p = 0.5 \).

The margin of error \( E \) is given as \( 0.025 \).

Using the formula for the required sample size \( n \):

\[ n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \]

Substituting the values:

\[ n = \frac{(2.807)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.025)^2} \]

Calculating this gives:

\[ n \approx 3151.6996 \]

Since the sample size must be a whole number, we round up:

\[ n = 3152 \]

Final Answer