An IQ test is designed so that the mean is 100 and the standard deviation is 8 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 $2 I Q$ points of the true mean. Assume that $\sigma=8$ 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 $\square$. (Round up to the nearest integer.)

Step 1 ：The problem is asking for 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 2 IQ points of the true mean. The standard deviation is given as 8.

Step 2 ：To solve this problem, we can use the formula for the sample size in a confidence interval estimation, which is: \(n = (Z*σ/E)^2\) where: n is the sample size, Z is the Z-score (which corresponds to the desired confidence level, in this case, 99%), σ is the standard deviation (which is 8 in this case), E is the margin of error (which is 2 in this case).

Step 3 ：We can find the Z-score for a 99% confidence level from a standard normal distribution table. The Z-score for a 99% confidence level is approximately 2.5758293035489004.

Step 4 ：Substitute the values into the formula: \(n = (2.5758293035489004*8/2)^2\)

Step 5 ：After calculating the sample size, we should round up to the nearest integer because the sample size should be a whole number. The calculated sample size is approximately 107.

Step 6 ：Final Answer: The required sample size is \(\boxed{107}\).