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 exceeded $3.5 million.

Question

Quizplus · Accepted Answer

We can use the Central Limit Theorem to approximate the distribution of the sample mean. Since the population standard deviation is known and the sample size is greater than 30, we can assume that the sample mean follows a normal distribution with mean = $3.26 million and standard deviation = $1.2 million/sqrt(100) = $0.12 million.

Now we need to find the probability that the sample mean exceeds $3.5 million. We can standardize the sample mean using the formula:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (3.5 - 3.26) / (1.2 / sqrt(100)) = 2.08

Using a standard normal table or calculator, we find that the probability of a standard normal variable exceeding 2.08 is approximately 0.0228.

Therefore, the approximate probability that the mean salary of the 100 players exceeded $3.5 million is 0.0228, or choice B.

Major League Baseball Salaries Averaged $3