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

Charles is a notoriously bad student who never studies for hisquizzes. 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 mightassume that he is going to guess on eachof the four questions.

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

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

X	P(X)	X · P(X)	X2 ·P(X)
0
1
2
3
4

Part 2. (4 points) Using the table youconstructed, find the expected value, ?, and the standarddeviation, ?. Please show your work.

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

Part 4. (2 points) Alternatively, astatistician might consider an outcome to be “unusual” if it ismore than two standard deviations away from the mean. Based on thiscriterion, should we be surprised if Charles guesses three or morequestions correctly? Please explain briefly.

Part  5. (2 points) Suppose Charleshappens to get three out of four questions correct. When computingthe probabilities in problem 1, we assumed that Charles wasguessing on every question; based on your answers to problems 3 and4, do you think our assumption is plausible? What is an alternativeexplanation for Charles’s performance?