The average ticket price for a Spring Training baseball game is $29.89, with a standard deviation of $5.28. In a random sample of 40 Spring Training tickets, find the probability that the mean ticket price exceeds $33

0.01% probability that the mean ticket price exceeds $33

Step-by-step explanation:

To solve this question, we need 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 normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

Find the probability that the mean ticket price exceeds $33

This is 1 subtracted by the pvalue of Z when X = 33. So

By the Central Limit Theorem

has a pvalue of 0.9999

1 - 0.9999 = 0.0001

0.01% probability that the mean ticket price exceeds $33