Description:

The Office of Financial Aid at the University of Michigan currently quotes the cost of textbooks for undergraduate students per semester to be $524 (Links to an external site.).  Administrators are interested in verifying if their estimate is accurate or if this figure needs to be updated before the start of the next school year.  They know that in order to obtain a more realistic estimate they need to rely on student data, so they have turned to various organizations on campus for help. 

The data: three different samples.

Two organizations on campus took different random samples of undergraduate students at UM and for each student recording the cost of textbooks during the Winter 21 semester.  Furthermore, financial aid administrators are aware that Stats 250 gathered data on the cost of textbooks for all students currently enrolled in the course.  Heres a brief summary of the three different random samples:

Table 1: Numerical summaries for 3 different samples of UM undergraduate students.

Organization

Mean

Standard Deviation

Sample Size

Sample A

$346

$39

25

Sample B

$286

$47

200

Stats 250 (you have been provided with this data to explore using RStudio (Links to an external site.))

X

Y

Z

The results from the three samples show very different estimates for the average cost of textbooks for one semester for undergraduate students at UM.  The Office of Financial Aid administrators are having a hard time making sense of the information provided and have hired you as a consultant. 

Your Task:

Write a 400 – 600 word memorandum (according to Purdue Owl Guidelines (Links to an external site.)) to the Office of Financial Aid administrators explaining how the process of sampling and the nature of sampling distributions has led to different estimates for the average cost of textbooks for one semester and provide a recommendation for the office of financial aid administrators given the results of the three samples.

Your memorandum must include responses to the following parts and all associated questions:

A brief introduction to the exploration and the third sample:
What population is the Office of Financial Aid administrators interested in?
Is the Stats 250 data representative for this population? Justify your answer
Explore the Stats 250 data using RStudio (include a histogram, a complete description of the distribution of cost of textbooks and numerical summaries to complete Table 1).
What factors involving the winter 21 semester could be affecting the response of interest (cost of textbook)?
All about Statistics:
You notice that these three estimates of mean textbook costs are different from each other.  Is that concerning to you?  Why or why not? 
Then, recall the idea of the sampling distribution, explored in Lab 5. Discuss how all of the features of the sampling distribution are affected by the sample size. Features could stay the same, change, or depend on the situation.
Why is a larger sample more accurate compared to a smaller sample?
Given these three estimates, which one is more reliable? Why? 
Recommendations for the Office of Financial Aid administrators:
The Office of Financial Aid has historical data that indicates the standard deviation of textbook costs is steady from semester to semester at $287.

If the true textbook cost is $524, what is the z-score and corresponding probability that youd get a sample with a mean like the one we observed in Sample A or less?  Provide an interpretation for your z-score.  (You could replicate this process for Sample B and for the Stats 250 data, although we arent asking for this here).
Using this probability, does the reported value of $524 seem reasonable?
Should the administrators adjust the value posted on the website?  Justify your recommendation using the statistical support provided in your memo.