Suppose your boss wants you to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the value of the population proportion. You would like to be $99.5 \%$ confident that your estimate is within $1 \%(0.01)$ of the true population proportion. What is the minimum sample size required?

IMPORTANT: Use a critical value that you found with your calculator (not from a table), and round it to 3 places after the decimal point before you plug it into a formula and perform your calculations. Do not round-off any other intermediate results.

Step 1 ：First, we need to define the desired level of confidence, the margin of error, and the estimated population proportion. In this case, the desired level of confidence is \(99.5\%\), the margin of error is \(1\%\) or \(0.01\), and since we have no reasonable estimate for the value of the population proportion, we will use \(0.5\) as a conservative estimate.

Step 2 ：Next, we need to find the z-score that corresponds to the desired level of confidence. The z-score is a measure of how many standard deviations an element is from the mean. We can find the z-score using the formula \(Z = \text{{stats.norm.ppf}}((1 + \text{{confidence_level}}) / 2)\). For a confidence level of \(99.5\%\), the z-score is approximately \(2.807\).

Step 3 ：Then, we calculate the minimum sample size required using the formula \(n = (Z^2 \times p \times (1 - p)) / E^2\). Substituting the values we have, we get \(n = (2.807^2 \times 0.5 \times (1 - 0.5)) / 0.01^2\).

Step 4 ：Since we can't have a fraction of a sample, we round up the result to the nearest whole number. The minimum sample size required is therefore \(\boxed{19699}\).