Question

A study finds that the number of gallons of water used each month per household in a

residential neighborhood are normally distributed with a mean of 470 gallons and a
standard deviation of 50 gallons. A household is randomly selected. What is the
probability that the household uses 480 gallons of water or less per month?

Answer 1

0.5793 = 57.93% probability that the household uses 480 gallons of water or less per month

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 470 gallons and standard deviation of 50 gallons.

This means that
`\mu = 470, \sigma = 50`

What is the probability that the household uses 480 gallons of water or less per month?

This is the pvalue of Z when X = 480. So

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

`Z = (480 - 470)/(50)`

`Z = 0.2`

`Z = 0.2` has a pvalue of 0.5793

0.5793 = 57.93% probability that the household uses 480 gallons of water or less per month