Question

Salaries differ greatly between faculty members at public universities in the United States. Full professors (who have tenure and do research) earn an average of $125,572 per year with a standard deviation of $18,328. Associate professors (who are working towards tenure and do research) earn an average of $90,101 per year with a standard deviation of $12,741. Lecturers (whose primary responsibility is teaching and do not earn tenure) earn an average of $59,707 per year with a standard deviation of $7,386. Assume all three populations are normally distributed.
a. What is the probability that a randomly selected full professor makes less than $100,000 per year?

Answer 1

We standardize the $100,000 as a z score of

z = (100,000 - 125572)/18328 = -1.395

That's 1.4 standard deviations below the mean.   We look up the area of the standard normal from -infinity to -1.40 standard deviations and find

P(salary < $100,00) = 0.08076 = 8%