Question

A bottler of drinking water fills plastic bottles with a mean volume of 1,007 milliliters (mL) and standard deviation The fill volumes are normally distributed. What proportion of bottles have volumes less than 1,007 mL?

Answer 1

0.5 = 50% of bottles have volumes less than 1,007 mL

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 question, we have that:

`\mu = 1007`

What proportion of bottles have volumes less than 1,007 mL?

This is the pvalue of Z when X = 1007. So

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

`Z = (1007 - 1007)/(\sigma)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

0.5 = 50% of bottles have volumes less than 1,007 mL