Question

The number of gallons of carbonated soft drink consumed per person annually is normally distributed with mean 47.5 and standard deviation 3.5.The probability that a randomly selected person consumes between 45 and 50 gallons of carbonated soft drink per year is about:

Answer 1

`P(45<X<50)=P((45-\mu)/(\sigma)<(X-\mu)/(\sigma)<(50-\mu)/(\sigma))=P((45-47.5)/(3.5)<Z<(50-47.5)/(3.5))=P(-0.714<z<0.714)`

And we can find this probability with the following difference:

`P(-0.714<z<0.714)=P(z<0.714)-P(z<-0.714)`

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.

`P(-0.714<z<0.714)=P(z<0.714)-P(z<-0.714)=0.762-0.238=0.524`

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 number of gallons of a population, and for this case we know the distribution for X is given by:

`X \sim N(47.5,3.5)`

Where
`\mu=47.5` and
`\sigma=3.5`

We are interested on this probability

`P(45<X<50)`

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(45<X<50)=P((45-\mu)/(\sigma)<(X-\mu)/(\sigma)<(50-\mu)/(\sigma))=P((45-47.5)/(3.5)<Z<(50-47.5)/(3.5))=P(-0.714<z<0.714)`

And we can find this probability with the following difference:

`P(-0.714<z<0.714)=P(z<0.714)-P(z<-0.714)`

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.

`P(-0.714<z<0.714)=P(z<0.714)-P(z<-0.714)=0.762-0.238=0.524`