An IQ test is designed so that the mean is $100$ and the standard deviation is $19$ for the population of normal adults Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with $99\%$ confidence that the sample mean is within $5IQ$ points of the true mean. Assume that $=19$ and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.

The required sample size is $($Round up to the nearest integer.$)$