Question

A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the actual capacity of a randomly selected tank has a distribution that is approximately Normal with a mean of 15.0 gallons and a standard deviation of 0.15 gallons. The manufacturer of this gasoline tank considers the largest 2% of these tanks too large to put on the market. How large does a tank have to be to be considered too large

Answer 1

Tanks of 15.3081 gallons and larger are considered too large.

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 = 15, \sigma = 0.15`

How large does a tank have to be to be considered too large

largest 2%, so at least the 98th percentile.

The 98th percentile is X when Z = 0.98. So it is X when Z = 2.054.

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

`2.054 = (X - 15)/(0.15)`

`X - 15 = 2.054*0.15`

`X = 15.3081`

Tanks of 15.3081 gallons and larger are considered too large.