Question

Directions: Answer the following questions. Round probabilities to four digits after the decimal. For full credit, you will need to show your work justifying how you determined numerical values. You may consider adding additional blank space to this document, printing, filling out by hand, and then uploading a scan or pictures of your answers. You may also re-write these questions on a separate sheet of paper and use as much space as you need to answer the questions.

Charles is a notoriously bad student who never studies for his quizzes. One day, his teacher gives a four question, multiple-choice quiz. Each question has five answer options (a, b, c, d, and e) and each question has only one correct answer option. Since Charles doesn’t study, we might assume that he is going to guess on each of the four questions.

Let X be a discrete random variable representing the number of questions Charles may correctly answer.

Part 1. (5 points) Complete the following probability distribution table:

X	P(X)	X · P(X)	X2 ·P(X)
0
1
2
3
4

Part 2. (4 points) Using the table you constructed, find the expected value, μ, and the standard deviation, σ. Please show your work.

Part 3. (2 points) A statistician might consider an outcome with probability less than 0.05 to be “unusual”. Based on this criterion, should we be surprised if Charles guesses three or more questions correctly? Please explain briefly.

Part 4. (2 points) Alternatively, a statistician might consider an outcome to be “unusual” if it is more than two standard deviations away from the mean. Based on this criterion, should we be surprised if Charles guesses three or more questions correctly? Please explain briefly.

Part  5. (2 points) Suppose Charles happens to get three out of four questions correct. When computing the probabilities in problem 1, we assumed that Charles was guessing on every question; based on your answers to problems 3 and 4, do you think our assumption is plausible? What is an alternative explanation for Charles’s performance?

Answer 1