Question 3
You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately $\sigma=38.3$. You would like to be $99 \%$ confident that your estimate is within 1.5 of the true population mean. How large of a sample size is required?
[Do not round mid-calculation. However, use a critical value rounded to three decimal places - this is important for the system to check answers correctly.]
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Step 1 ：Given that the population standard deviation \(\sigma = 38.3\), the desired margin of error \(E = 1.5\), and the z-score corresponding to the 99% confidence level \(Z_{\alpha/2} = 2.576\).

Step 2 ：We can calculate the required sample size using the formula for the sample size in estimating a population mean with a certain level of confidence: \(n = \left(\frac{Z_{\alpha/2} * \sigma}{E}\right)^2\).

Step 3 ：Substitute the given values into the formula: \(n = \left(\frac{2.576 * 38.3}{1.5}\right)^2\).

Step 4 ：Solving the equation gives us \(n = 4327\).

Step 5 ：So, the required sample size is \(\boxed{4327}\).