Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000. Using Chebyshev's inequality, the percentage of salaries falling between $68,000 and $92,000 is at least ________.
A) 68%
B) 65%
C) 75%
D) 95%

Question

Quizplus · Accepted Answer

Chebyshev's inequality states that at least 1 - (1/k^2) of the data will fall within k standard deviations of the mean, where k is any number greater than 1. To find the percentage of salaries falling between $68,000 and $92,000, we need to find how many standard deviations away from the mean those values are: $68,000 - $80,000 = -$12,000 -$12,000/$6,000 = -2 $92,000 - $80,000 = $12,000 $12,000/$6,000 = 2 Therefore, the range from $68,000 to $92,000 is 2 standard deviations away from the mean. Using the inequality, we can say: P(|X - μ| < kσ) >= 1 - (1/k^2) where X = salary of professors, μ = mean salary ($80,000), σ = standard deviation ($6,000), and k = 2 (since we want to find the range within 2 standard deviations from the mean). Plugging in the values, we get: P(|X - 80,000| < 2(6,000)) >= 1 - (1/2^2) P(68,000 < X < 92,000) >= 3/4 So we can say with at least 75% confidence that salaries fall between $68,000 and $92,000. Therefore, the answer is C.

Professors at a Local University Earn an Average Salary of $80,000