Question

Suppose that the number of gallons of milk sold per day at a local supermarket are normally distributed with mean and standard deviation of 442.8 and 20.6, respectively. What is the probability that on a given day the supermarket will sell between 410 and 424 gallons of milk?

Answer 1

12.55% probability that on a given day the supermarket will sell between 410 and 424 gallons of milk.

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 = 442.8, \sigma = 20.6`

What is the probability that on a given day the supermarket will sell between 410 and 424 gallons of milk?

This is the pvalue of Z when X = 424 subtracted by the pvalue of Z when X = 410.

X = 424

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

`Z = (424 - 442.8)/(20.6)`

`Z = -0.91`

`Z = -0.91` has a pvalue of 0.1814.

X = 410

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

`Z = (410 - 442.8)/(20.6)`

`Z = -1.59`

`Z = -1.59` has a pvalue of 0.0559.

So there is a 0.1814 - 0.0559 = 0.1255 = 12.55% probability that on a given day the supermarket will sell between 410 and 424 gallons of milk.