One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.8 million. Suppose a sample of 400 major league players was taken. Find the
Approximate probability that the average salary of the 400 players that year exceeded $1.1 million.
A)approximately 0
B)approximately 1
C).2357
D).7357

Question

Quizplus · Accepted Answer

Given the large sample size (n = 400), we can use the Central Limit Theorem to approximate the distribution of the sample mean. The standard error (SE) of the sample mean is the standard deviation divided by the square root of the sample size, SE = 0.8 million / sqrt(400) = 0.8 / 20 = 0.04 million. The Z-score for a mean of $1.1 million is (1.1 - 1.6) / 0.04 = -12.5. This Z-score is extremely low, indicating that the probability of the sample mean being less than $1.1 million is almost 0, so the probability of it being more than $1.1 million is approximately 1.

One Year, the Distribution of Salaries for Professional Sports Players