Question

In Melanie's Styling Salon, the time to complete a simple haircut is normally distributed with a mean of 25 minutes and a standard deviation of 4 minutes.

The slowest quartile of customers will require longer than how many minutes for a simple haircut?



a) 3(n+1)/4 minutes

b) 26 minutes

c) 25.7 minutes

d) 27.7 minutes

Answer 1

`z=0.674<(a-25)/(4)`

And if we solve for a we got

`a=25 +0.674*4=27.7`

So the value of height that separates the bottom 75% of data from the top 25% is 27.7 minutes.

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 time to complete a simple haircut of a population, and for this case we know the distribution for X is given by:

`X \sim N(25,4)`

Where
`\mu=25` and
`\sigma=4`

We are interested in the slowest quartile or the first quartile

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

`P(X>a)=0.25` (a)

`P(X<a)=0.75` (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 0.75 of the area on the left and 0.25 of the area on the right it's z=0.674. On this case P(Z<0.674)=0.75 and P(z>0.674)=0.25

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

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.75`

`P(z<(a-\mu)/(\sigma))=0.75`

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

`z=0.674<(a-25)/(4)`

And if we solve for a we got

`a=25 +0.674*4=27.7`

So the value of height that separates the bottom 75% of data from the top 25% is 27.7 minutes.