One year, professional sports players salaries averaged 1.6 million with a standard deviation of .7 million. Suppose a sample of 100 major league players was taken. Find the approximate probability that the average salary of the 100 players exceeded 1.1 million.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the approximate probability that the average salary of the 100 players exceeded 1.1 million, we can use the Central Limit Theorem (CLT) and approximate the distribution of sample means as a normal distribution.
The Central Limit Theorem states that regardless of the shape of the population distribution, as the sample size increases, the distribution of the sample means will become approximately normally distributed.
In this case, the mean salary of professional sports players is 1.6 million, and the standard deviation is 0.7 million. Since we have a large sample size (100 players), we can assume that the distribution of sample means will be approximately normal.
To calculate the probability, we need to standardize the value of 1.1 million using the formula for z-scores:
z = (x - μ) / (σ / sqrt(n))
Where:
x is the value we want to standardize (1.1 million),
μ is the population mean (1.6 million),
σ is the population standard deviation (0.7 million),
n is the sample size (100).
Now we can calculate the z-score:
z = (1.1 - 1.6) / (0.7 / sqrt(100))
= (-0.5) / (0.7 / 10)
= -0.5 / 0.07
= -7.14 (rounded to two decimal places)
Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability associated with that z-score.
Looking up the z-score of -7.14 in a standard normal distribution table, we find that the probability (or area under the curve) is essentially zero, as the extreme negative z-score indicates an observation far below the mean.
Therefore, the approximate probability that the average salary of the 100 players exceeded 1.1 million is close to zero.