Question

The amount of cereal in a box is normal with mean 16.5 ounces. If the packager is required to fill at least 90% of the cereal boxes with 16 or more ounces of cereal, what is the largest standard deviation for the amount of cereal in a box?

Answer 1

The largest standard deviation for the amount of cereal in a box is 0.3906.

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 = 16`

The z-score of X = 16 has a pvalue of 0.1. So it is
`Z = -1.28`. Now we have to find the value of
`\sigma`.

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

`-1.28 = (16 - 16.5)/(\sigma)`

`-1.28\sigma = -0.5`

`\sigma = 0.3906`

The largest standard deviation for the amount of cereal in a box is 0.3906.