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

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:

[tex]X \sim N(25,4)[/tex]

Where [tex]\mu=25[/tex] and [tex]\sigma=4[/tex]

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:

[tex]P(X>a)=0.25[/tex] (a)

[tex]P(X<a)=0.75[/tex] (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:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.75[/tex]

[tex]P(z<\frac{a-\mu}{\sigma})=0.75[/tex]

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

[tex]z=0.674<\frac{a-25}{4}[/tex]

And if we solve for a we got

[tex]a=25 +0.674*4=27.7[/tex]

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