Question

The amount of money a Chili's bartender makes in tips on any given night is normally distributed with a mean of $125 and a standard deviation of $4.80. If 36 Chili's bartenders are randomly selected, find the probability that their average earned tippings is less than $127.

Answer 1

99.38% probability that their average earned tippings is less than $127.

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
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (127 - 125)/(0.8)`

`Z = 2.5`

`Z = 2.5` has a pvalue of 0.9938

99.38% probability that their average earned tippings is less than $127.