Probability Calculation for Professional Sports Players' Salaries
Understanding Probability Calculation for Professional Sports Players' Salaries
Central Limit Theorem: The Central Limit Theorem is a fundamental concept in statistics that states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
Calculating Standard Error (SE) of the Sample Mean:
Standard Error (SE): The standard error of the sample mean is a measure of the dispersion of sample means around the population mean. In this case, we calculated the standard error as $0.035 million, indicating the variability in the average salaries of the 400 players.
Interpreting Z-Score and Probability:
Z-Score: The z-score measures how many standard deviations a data point is from the mean. In this scenario, the z-score of -17.14 indicates that the sample mean of $1.1 million is significantly below the population mean of $1.7 million.
Probability Interpretation: The probability of the average salary of the 400 players exceeding $1.1 million is virtually certain (close to 100%) due to the extremely low z-score and the corresponding area to the right of it in the normal distribution curve.