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

Answer:

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

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

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 125, \sigma = 4.8, s = \frac{4.8}{\sqrt{36}} = 0.8[/ex]

If 36 Chili's bartenders are randomly selected, find the probability that their average earned tippings is less than 127.

This is the pvalue of Z when X = 127. So

[tex]Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{127 - 125}{0.8}$

$Z = 2.5$

$Z = 2.5$ has a pvalue of 0.9938

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