Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a certain year in the past.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.

Question

Quizplus · Accepted Answer

We need to use the central limit theorem here, as the sample size is large enough. The mean of a sample of size 100 will follow a normal distribution with mean= $3.26 million, and standard deviation= $1.2 million / sqrt(100) = $0.12 million. We want to find the probability that this sample mean is no more than $3.0 million, which can be written as: 
P(x̄ ≤ $3.0) = P((x̄-μ)/(σ/√n) ≤ (3.0-3.26)/(1.2/√100))
= P(Z ≤ -2.167) 
Using a standard normal distribution table, we find that the probability of Z being less than or equal to -2.167 is approximately 0.0151. Therefore, the probability that the mean salary of the 100 players was no more than $3.0 million is approximately 0.0151.

Major League Baseball Salaries Averaged $3