An IQ test is designed so that the mean is 100 and the standard deviation is 10 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such can be said with \( 95 \% \) confidence that the sample mean is within 2 IQ points of the true mean. Assume that \( \sigma=10 \) and determine the required sample size using technology. Then determine if this is a reaso sample size for a real world calculation. The required sample size is (Round up to the nearest integer)

To find the necessary sample size, you can use the formula: \[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 \] Where: - \( n \) = required sample size - \( Z \) = z-value corresponding to the desired confidence level (for \( 95\% \), \( Z \approx 1.96 \)) - \( \sigma \) = population standard deviation (given as \( 10 \)) - \( E \) = margin of error (the desired precision, which is \( 2 \) IQ points) Plugging in the values: \[ n = \left(\frac{1.96 \cdot 10}{2}\right)^2 = \left(9.8\right)^2 = 96.04 \] Since you always round up when determining sample size, the required sample size is \( 97 \). This sample size is quite reasonable, especially in an academic setting like statistics students. Gathering 97 responses allows for a robust estimation to capture the reality of their IQ scores, maintaining the integrity of your findings while also being manageable in terms of logistics!