3.68 Students in a business statistics class were asked what grade they expected in the course and whether they worked on additional problems beyond those assigned by the instructor. The following table gives proportions of students in each of eight joint classifications.

a. Find the probability that a randomly chosen student from this class worked on additional problems.

b. Find the probability that a randomly chosen student from this class expects an A.

c. Find the probability that a randomly chosen student who worked on additional problems expects an A.

d. Find the probability that a randomly chosen student who expects an A worked on additional problems.

e. Find the probability that a randomly chosen student who worked on additional problems expects a grade below B.

f. Are “worked additional problems” and “expected grade” statistically independent?

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Worked Expected Grade Problems A B C Below C Yes 0.12 0.06 0.12 0.02 No 0.13 0.21 0.26 0.08