The average score of all pro golfers for a particular course has a mean of 70 and a standard deviation of 3.0. Suppose 36 pro golfers played the course today. Find the probability that the average score of the 36 pro golfers exceeded 71.
To find the probability that the average score of the 36 pro golfers exceeded 71, we can use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means will be approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
First, we need to find the standard error of the sample mean:
Standard error = standard deviation / sqrt(sample size)
Standard error = 3.0 / sqrt(36)
Standard error = 3.0 / 6
Standard error = 0.5
Next, we calculate the z-score of 71:
z = (X - μ) / SE
z = (71 - 70) / 0.5
z = 1 / 0.5
z = 2
Now, we find the probability that the average score exceeded 71 by using the z-score:
P(X > 71) = P(Z > 2)
Using a standard normal distribution table, we can find that the probability of Z exceeding 2 is approximately 0.0228 or 2.28%.
Therefore, the probability that the average score of the 36 pro golfers exceeded 71 is approximately 2.28%.