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 exceeded $3.5 million.
A)approximately 0
B)0.0228
C)0.9772
D)approximately 1

Question

Quizplus · Accepted Answer

We can use the central limit theorem to approximate that the sample mean, X̄ , follows a normal distribution with mean μ =$3.26 million and standard deviation σ/√n= $1.2 million/√100= $0.12 million. We want to find P(X̄ > $3.5 million).

z=(X̄-μ)/σ√n=(3.5-3.26)/(1.2/√100)=1.67

Using a normal distribution table, we get P(Z>1.67)=0.0475, which is the approximate probability of the sample mean exceeding $3.5 million. However, this is the probability of a single sample mean exceeding $3.5 million, not the probability that the population mean exceeds $3.5 million. Therefore, we cannot choose choice D. Since this is a one-tailed test (we are only interested in the probability that the sample mean is greater than $3.5 million), we use the right tail probability, which is 0.0475. Therefore, the best choice is B.

Major League Baseball Salaries Averaged $3