The Math 122 Midterm Exam is coming up. Suppose the exam scores are normally distributed with a population mean of 71.4% and a standard deviation of 14.5%.

Let's first create a simulation to observe the expected results for a class of Math 122 students In Excel, create 25 random samples of 19 students each. This means you should have 19 entries in each column, and you should be using columns A-Y. If you need a refresher for creating a random sample that is normally distributed, you can review the Technology Corner from Module 2.

After creating your random samples, copy all the numbers then use the "Paste Values" option in Google Sheets to lock the numbers in place. Save your file as an .xlsx, then attach it here (File >> Download >> Microsoft Excel):

Now find the mean of each sample.

What is the highest mean?

What is the lowest mean?

Note: While there are no points associated with the attachment or the highest/lowest mean, points will be deducted for not completing this portion or doing it incorrectly. These should be used to help you understand the remainder of the problem.

What is the probability of a student getting a score of 90% or better?  (Round to four decimal places.)

What is the probability of a class of 24 students having a mean of 90% or better?  (Round to six decimal places.)

Explain why the answers to these two questions are drastically different. Your explanation should include each of the following:

references to your simulation
references to your calculations
common sense explanation