IQ test is designed so that the mean is 100 and the standard deviation is 12 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it be said with \( 99 \% \) confidence that the sample mean is within 7 IQ points of the true mean. Assume that \( \sigma=12 \) and determine the required sample size using technology. Then determine if this is a reasonable nple size for a real world calculation. e required sample size is \( \square \). (Round up to the nearest integer.)

To determine the required sample size for estimating the mean IQ score, we can use the formula for sample size \( n \): \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \] Where: - \( Z \) is the z-value corresponding to the desired confidence level (for 99% confidence, \( Z \approx 2.576 \)). - \( \sigma \) is the population standard deviation (given as 12). - \( E \) is the margin of error (given as 7). Plugging in the values: \[ n = \left( \frac{2.576 \cdot 12}{7} \right)^2 \approx \left( \frac{30.912}{7} \right)^2 \approx (4.413)^2 \approx 19.5 \] Since the sample size must be a whole number, we round up to the nearest integer, giving us a required sample size of \( 20 \). Now, regarding the practicality of a sample size of 20, it can be quite reasonable in real-world scenarios, especially in academic settings. It's small enough to manage easily but large enough to yield statistically significant results, assuming the data meets the necessary assumptions for normalcy and independence. Managing logistics, obtaining responses or testing the participants, is viable with this size. Sample Size = \( 20 \)