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. for a simple haircut, the middle 90 percent of the customers will require:

Answer 1

They will require between 18.42 and 31.58 minutes.

We want the middle 90%.  The two probability values associated with this would be 0.95 and 0.05; this would leave 5% below and 5% above, giving us the middle 90%.

Using a z-table (http://www.z-table.com) we see that the z-score associated with an area to the left of 0.05 is between -1.64 and -1.65; since it is equally distant from both we will use -1.645.

The z-score associated with an area of the left of 0.95 is between 1.64 and 1.65; since it is equally distant from both we will use 1.645.

The formula for a z-score is

z = (X-μ)/σ
-1.645 = (X-25)/4

Multiplying by 4 on both sides,
-1.645(4) = X-25
-6.58 = X-25

Adding 25 to both sides,
-6.58+25 = X
18.42 = X

For the upper bound,
1.645 = (X-25)/4

Multiplying both sides by 4,
1.645(4) = X-25
6.58 = X-25

Adding 25 to both sides,
6.58+25 = X
31.58 = X

The times are between 18.42 and 31.58 minutes.