Historically, the average score of PGA golfers for one round is 73.9 with a standard deviation of 1.1. A random sample of 99 golfers is taken. What is the probability that the sample mean is between 73.9 and 74.03

38.10% probability that the sample mean is between 73.9 and 74.03.

Step-by-step explanation:

To solve this question, it is important to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

What is the probability that the sample mean is between 73.9 and 74.03

This is the pvalue of Z when X = 74.03 subtracted by the pvalue of Z when X = 73.9. So

X = 74.03

By the Central Limit Theorem

has a pvalue of 0.8810.

X = 73.9

has a pvalue of 0.5

0.8810 - 0.5 = 0.3810

38.10% probability that the sample mean is between 73.9 and 74.03.

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: