Question

f we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes. You should use the finite population correction factor for this problem. True False

Answer 1

`P(X<3)=P((X-\mu)/(\sigma)<(3-\mu)/(\sigma))=P(Z<(3-3.5)/(1))=P(z<-0.5)`

And we can find this probability with the normal standard table or excel and we got:

`P(z<-0.5)=0.3085`

And for this case is not neccesary applt the finite population correction factor since the distribution for the random variable is assumed known and normal

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

`X \sim N(3.5,1)`

Where
`\mu=3.5` and
`\sigma=1`

We are interested on this probability

`P(X<3)`

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<3)=P((X-\mu)/(\sigma)<(3-\mu)/(\sigma))=P(Z<(3-3.5)/(1))=P(z<-0.5)`

And we can find this probability with the normal standard table or excel and we got:

`P(z<-0.5)=0.3085`

And for this case is not neccesary applt the finite population correction factor since the distribution for the random variable is assumed known and normal