Question

The amount of cereal dispensed into "12-ounce" boxes of Captain Crisp cereal is normally distributed with mean 12.09 ounces and standard deviation 0.12 ounces. a. What proportion of boxes are "underfilled"? That is, what is the probability that the amount dispensed into a box is less than 12 ounces? Give your answer to 4 decimal places.

Answer 1

There is a 22.66% probability that the amount dispensed into a box is less than 12 ounces.

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)`

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.

In this problem:

The amount of cereal dispensed into "12-ounce" boxes of Captain Crisp cereal is normally distributed with mean 12.09 ounces and standard deviation 0.12 ounces, so
`\mu = 12.09, \sigma = 0.12`.

That is, what is the probability that the amount dispensed into a box is less than 12 ounces?

This is the pvalue of Z when
`X = 12`.

So:

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

`Z = (12 - 12.09)/(0.12)`

`Z = -0.75`

`Z = -0.75` has a pvalue of 0.2266.

This means that there is a 22.66% probability that the amount dispensed into a box is less than 12 ounces.