Questions: How large a sample should be selected to provide a 95% confidence interval with a margin of error of 57? Assume that the population standard deviation is 50. Round your answer to the next whole number.

Solution Steps

To find the Z-score corresponding to a \(95\%\) confidence level, we use the formula: \[ Z = \text{PPF}\left(1 - \frac{1 - 0.95}{2}\right) = \text{PPF}(0.975) = 1.96 \]

Using the Z-score, population standard deviation \(\sigma = 50\), and the desired margin of error \(E = 57\), we can calculate the required sample size \(n\) using the formula: \[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 = \left(\frac{1.96 \cdot 50}{57}\right)^2 \]

After calculating the sample size, we find: \[ n \approx 2.9559 \] Rounding this value to the next whole number gives us the final required sample size.

Final Answer