Probability Calculation for Professional Sports Players' Salaries

What is the approximate probability that the average salary of 400 professional sports players exceeds $1.1 million? The approximate probability that the average salary of 400 professional sports players exceeds $1.1 million can be calculated using the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough. Given: - Population mean (μ) = $1.7 million - Population standard deviation (σ) = $0.7 million - Sample size (n) = 400 players First, we calculate the standard error (SE) of the sample mean: SE = σ / √n = $0.7 million / √400 = $0.035 million Next, we find the z-score for the sample mean of $1.1 million: z = ($1.1 million - $1.7 million) / $0.035 million = -$17.14 Since the z-score is negative and we are interested in the probability that the average salary exceeds $1.1 million, we look at the area to the right of the z-score. The area to the right of such a large negative z-score is approximately 1 (or almost certain). Therefore, the approximate probability that the average salary of the 400 players exceeds $1.1 million is very close to 1, or 100%.

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.

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