Question

The daily demand for milk (in gallons) at Gillis Grocery is N(1,000,100). How many gallons must be in stock at the beginning of the day if Gillis is to have only a 5% chance of running out of milk by the end of the day?

Answer 1

1164.5 gallons of milk.

Explanation:

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.

In this problem, we have that:

`\mu = 1000, \sigma = 100`

How many gallons must be in stock at the beginning of the day if Gillis is to have only a 5% chance of running out of milk by the end of the day?

This is the value of X when Z has a pvalue of 1-0.05 = 0.95. So it is X when
`Z = 1.645`. So

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

`1.645 = (X - 1000)/(100)`

`X - 1000 = 1.645*100`

`X = 1164.5`

1164.5 gallons of milk.