Question

Parking concerns ~ Over the years, parking has become an issue during spring at MSU as more people bring cars to campus. The waiting time to find a parking spot at Wells Hall parking lot is normally distributed with an average of 10 minutes and a standard deviation of 3.6 minutes during peak hours (10 am – 5 pm). What proportion of waiting times are between 8 and 12 minutes? Enter your answer to 4 decimal places.

Answer 1

`P(8<X<12)=P((8-\mu)/(\sigma)<(X-\mu)/(\sigma)<(12-\mu)/(\sigma))=P((8-10)/(3.6)<Z<(12-10)/(3.6))=P(-0.56<z<0.56)`

And we can find this probability with this difference:

`P(-0.56<z<0.56)=P(z<0.56)-P(z<-0.56)`

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

`P(-0.56<z<0.56)=P(z<0.56)-P(z<-0.56)=0.7123-0.2877=0.4246`

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

`X \sim N(10,3.6)`

Where
`\mu=10` and
`\sigma=3.6`

We are interested on this probability

`P(8<X<12)`

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(8<X<12)=P((8-\mu)/(\sigma)<(X-\mu)/(\sigma)<(12-\mu)/(\sigma))=P((8-10)/(3.6)<Z<(12-10)/(3.6))=P(-0.56<z<0.56)`

And we can find this probability with this difference:

`P(-0.56<z<0.56)=P(z<0.56)-P(z<-0.56)`

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

`P(-0.56<z<0.56)=P(z<0.56)-P(z<-0.56)=0.7123-0.2877=0.4246`