An economist needs to estimate the proportion of residents of Vero Beach that have earned a college degree. Determine the most conservative estimate of the sample size required to limit the margin of error to within 0.038 of the population proportion for a $99 \%$ confidence interval.

Round the solution up to the nefrest whole number.
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Step 1 ：Given values are: Z-score for 99% confidence level \(Z = 2.576\), margin of error \(E = 0.038\), and the most conservative estimate of the population proportion \(p = 0.5\).

Step 2 ：Calculate the sample size using the formula \(n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\).

Step 3 ：Substitute the given values into the formula: \(n = \frac{(2.576)^2 \cdot 0.5 \cdot (1-0.5)}{(0.038)^2}\).

Step 4 ：Calculate the value of \(n\), and round up to the nearest whole number to get \(n = 1149\).

Step 5 ：Final Answer: The most conservative estimate of the sample size required to limit the margin of error to within 0.038 of the population proportion for a 99% confidence interval is \(\boxed{1149}\).