The standard IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of all adults. We wish to find the sample size necessary to estimate the mean IQ score of statistics students. Suppose we want to be $96\mathrm{\%}$ confident that our sample mean is within $2\mathrm{IQ}$ points of the true mean. The mean for this population is clearly greater than 100 . The standard deviation for this population is probably less than 15 because it is a group with less variation than a group randomly selected from the general population; therefore, if we use $\sigma =15$, we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that $\sigma =15$ and determine the required sample size.
Answer:

Step 1 ：We are given that the population standard deviation (σ) is 15 and the margin of error (E) is 2 IQ points. We want to find the sample size (n) necessary to estimate the mean IQ score of statistics students with a 96% confidence level.

Step 2 ：We can use the formula for the sample size in a confidence interval estimation for a population mean, which is: $n=(Z\ast \sigma /E{)}^2$, where Z is the Z-score (which corresponds to the desired confidence level).

Step 3 ：We need to find the Z-score that corresponds to a 96% confidence level. The Z-score for a 96% confidence level is approximately 2.05 (you can find this value in a standard Z-table or use a statistical calculator).

Step 4 ：Substituting these values into the formula, we get: $n=(2.05\ast 15/2{)}^2$.

Step 5 ：Calculating the above expression, we find that the required sample size is approximately 237.

Step 6 ：Final Answer: The required sample size to estimate the mean IQ score of statistics students with a 96% confidence level and a margin of error of 2 IQ points is $\boxed{{\displaystyle 237}}$.