Question

the cost for a male haircut is normally distributed with a mean of $28.00 and a standard deviation of $2.50. use this information for problems 7 - 9.

1. what is the probability that a male haircut is less than $25.50? (lesson 23.2)(1 point)
2. what is the probability that a male haircut is more than $33.00? (lesson 23.2)(1 point)
3. what is the probability that the price of a haircut of a randomly selected male is between $23.00 and $33.00? (lesson 23.2) (1 point)

Explanation:

Step1: Calculate the z - score for problem 7

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\). Given \(\mu = 28\), \(\sigma=2.5\), and \(x = 25.5\).

$$z=\frac{25.5 - 28}{2.5}=\frac{- 2.5}{2.5}=-1$$

Looking at the standard normal distribution (the empirical rule or 68 - 95 - 99.7 rule), the area to the left of \(z=-1\) is \(16\%\) or \(0.16\)

Step2: Calculate the z - score for problem 8

Using the z - score formula \(z=\frac{x-\mu}{\sigma}\), with \(\mu = 28\), \(\sigma = 2.5\), and \(x = 33\)

$$z=\frac{33 - 28}{2.5}=\frac{5}{2.5}=2$$

The area to the right of \(z = 2\) is \(2.35\%+0.15\%=2.5\%\) or \(0.025\)

Step3: Calculate the z - scores for problem 9

For \(x = 23\):

$$z_1=\frac{23 - 28}{2.5}=\frac{-5}{2.5}=-2$$

For \(x = 33\):

$$z_2=\frac{33 - 28}{2.5}=2$$

The area between \(z=-2\) and \(z = 2\) is \(13.5\%+34\%+34\%+13.5\%=95\%\) or \(0.95\)

Answer:

1. The probability that a male haircut is less than \(\$25.50\) is \(0.16\)
2. The probability that a male haircut is more than \(\$33.00\) is \(0.025\)
3. The probability that the price of a haircut of a randomly selected male is between \(\$23.00\) and \(\$33.00\) is \(0.95\)