Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players was no more than $3.0 million.
A) Approximately 0
B) 0.0151
C) 0.9849
D) Approximately 1

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate the sampling distribution of the sample mean. Since n=100 is large, we can assume that the sample mean follows a normal distribution with mean μ = $3.26 million and standard deviation σ/√n = $1.2 million/√100 = $0.12 million.

To find the probability that the mean salary of the 100 players was no more than $3.0 million, we can standardize the sample mean:

z = (x̄ - μ)/(σ/√n) = ($3.0 million - $3.26 million)/($0.12 million) = -2.17

Using a standard normal distribution table, we find that the area to the left of z = -2.17 is approximately 0.0151. Therefore, the approximate probability that the mean salary of the 100 players was no more than $3.0 million is 0.0151, or choice B.

Major League Baseball Salaries Averaged $3