Question

Bottling cola A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean μ = 298ml and standard deviation σ = 3ml.

What is the probability that a randomly selected bottle contains less than 295 ml? Show your work.

Answer 1

`P(X<295)=P((X-\mu)/(\sigma)<(295-\mu)/(\sigma))=P(Z<(295-298)/(3))=P(Z<-1)`

And we can find this probability using the normal standard table or excel:

`P(Z<-1)=0.159`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the amount of ml of a population, and for this case we know the distribution for X is given by:

`X \sim N(298,3)`

Where
`\mu=298` and
`\sigma=3`

We are interested on this probability

`P(X<295)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(x-\mu)/(\sigma)`

If we apply this formula to our probability we got this:

`P(X<295)=P((X-\mu)/(\sigma)<(295-\mu)/(\sigma))=P(Z<(295-298)/(3))=P(Z<-1)`

And we can find this probability using the normal standard table or excel:

`P(Z<-1)=0.159`