Question

Suppose that the waiting time for a license plate renewal at a local office of a state motor vehicle department has been found to be normally distributed with a mean of 30 minutes and a standard deviation of 8 minutes. Suppose that in an effort to provide better service to the public, the director of the local office is permitted to provide discounts to those individuals whose waiting time exceeds a predetermined time. The director decides that 15 percent of the customers should receive this discount. What number of minutes do they need to wait to receive the discount?

A) 34.48
B) 21.68
C) 38.32
D) 25.52

Answer 1

`a=30 +1.04*8=38.32`

So the value of height that separates the bottom 85% of data from the top 15% is 38.32.

C) 38.32

Explanation:

1) 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".

2) Solution to the problem

Let X the random variable that represent the waiting time in minutes of a population, and for this case we know the distribution for X is given by:

`X \sim N(30,8)`

Where
`\mu=30` and
`\sigma=8`

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

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

We want to find a value a, such that we satisfy this condition:

`P(X>a)=0.15` (a)

`P(X<a)=0.85` (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.85 of the area on the left and 0.15 of the area on the right it's z=1.04. On this case P(Z<1.04)=0.85 and P(z>1.04)=0.15

If we use condition (b) from previous we have this:

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.85`

`P(z<(a-\mu)/(\sigma))=0.85`

But we know which value of z satisfy the previous equation so then we can do this:

`z=1.04<(a-30)/(8)`

And if we solve for a we got

`a=30 +1.04*8=38.32`

So the value of height that separates the bottom 85% of data from the top 15% is 38.32.

C) 38.32