Question

The Math 122 Midterm Exam is coming up. Suppose the exam scores are normally distributed with a population mean of 73.8% and a standard deviation of 11.3%.

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 23 students each. This means you should have 23 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 Excel to lock the numbers in place. Save your file, then attach it here:

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. This should be the theoretical probability that is calculated, NOT the empirical probability from the simulation.)

What is the probability of a class of 23 students having a mean of 90% or better?  (Round to six (6) decimal places.This should be the theoretical probability that is calculated, NOT the empirical probability from the simulation.)

Explain, in your own words, why the answers to these two questions are drastically different. Your explanation should include:

• references to your simulation
• references to your calculations
• common sense explanation

Answer 1

Sorry we cant attach excel sheet over here. I tried but its says # of character crossed 65000 limit.

X bar - Last row - row represents X bar of each sample

68.00 - lowest mean out of 25 samples

79.31 - highest mean out of 25 samples.

3. Probabilities are different because :

If you closely observe the simulation where we have obtained 23*25 = 575 data points from normal population with N (73.8, 11.3^2). there were few data points equal or more than 90. Here the variance of data points were 11.3^2.

But when we arranged the data into 25 samples of 23 data points each- sample mean of each of the sample tends to close to the population mean- as per central limit theorem. If you observe sample means of each sample i.e. X bar , all X bar's are close to population mean = 73.8 with very little standard deviation. (as per central limit theorem - as we increase the sample size data approaches to follow normal distribution and distribution becomes more narrow in shape, that means std. deviation decreases upon increasing sample size, here we have n = 23. which is very large as compared to n=1 in case of 1st part.

Since, std. deviation of sample means X bar's were very less , Probability of getting mean class score > 90% is nearly impossible as 90 is 6.88 standard deviations far away from mean of 73.8.