Question

The probability distribution of customers that walk into a coffee shop on any given day of the week is described by a Normal distribution with mean equal to 100 and standard deviation equal to 20. What is the probability that no more than 80 customers walk into the coffee shop next Monday

Answer 1

15.87% probability that no more than 80 customers walk into the coffee shop next Monday

Explanation:

Problems of normally distributed samples can be 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.

In this problem, we have that:

`\mu = 100, \sigma = 20`

What is the probability that no more than 80 customers walk into the coffee shop next Monday?

This is the pvalue of Z when X = 80. So

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

`Z = (80 - 100)/(20)`

`Z = -1`

`Z = -1` has a pvalue of 0.1587.

So there is a 15.87% probability that no more than 80 customers walk into the coffee shop next Monday