You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately $p=0.87$. You would like to be $98 \%$ confident that your esimate is within $3 \%$ of the true population proportion. How large of a sample size is required? Hint: Video $[$ [ $[+]$ \[ n= \]

Step 1 ：We want to obtain a sample to estimate a population proportion. Based on previous evidence, we believe the population proportion is approximately \(p=0.87\). We would like to be \(98 \%\) confident that our estimate is within \(3 \%\) of the true population proportion. We need to find out how large of a sample size is required.

Step 2 ：The sample size required can be calculated using the formula for sample size estimation for proportions. The formula is: \[n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\] where: \(n\) is the sample size, \(Z\) is the z-score for the desired confidence level, \(p\) is the estimated population proportion, and \(E\) is the desired margin of error.

Step 3 ：In this case, we have: \(p = 0.87\), \(E = 0.03\), and \(Z = 2.33\) (for a 98% confidence level).

Step 4 ：We can substitute these values into the formula to calculate the required sample size: \[n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2} = \frac{2.33^2 \cdot 0.87 \cdot (1-0.87)}{0.03^2}\]

Step 5 ：The calculated sample size is 683. This means that in order to be 98% confident that the estimate is within 3% of the true population proportion, a sample size of 683 is required.

Step 6 ：Final Answer: The required sample size is \(\boxed{683}\).