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:a) between 20.0 and 30.0 minutes.b) between 18.4 and 31.6 minutes.c) between 17.2 and 32.8 minutes.d) between 19.9 and 30.1 minutes.

Answer 1

Answer:

b) between 18.4 and 31.6 minutes.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 25, \sigma = 4$

Middle 90%

Lower bound: Value of X when Z has a pvalue of 0.5 - 0.9/2 = 0.05.

Upper bound: Value of X when Z has a pvalue of 0.5 + 0.9/2 = 0.95.

Lower bound:

X when Z = -1.645

$Z = \frac{X - \mu}{\sigma}$

$-1.645 = \frac{X - 25}{4}$

$X - 25 = -1.645*4$

$X = 18.42$

Upper bound

X when Z = 1.645

$1.645 = \frac{X - 25}{4}$

$X - 25 = 1.645*4$

$X = 31.58$

So the correct answer is:

b) between 18.4 and 31.6 minutes.

Answer 2

For a simple haircut, the middle 90 percent of the customers will require between 18.4 and 31.6 minutes.

So, Option (b) is correct.

Normal distribution is a continuous probability distribution where values lie in a symmetrical fashion mostly situated around the mean.

Here, we use Z score rule in normally distribution.

Let us assume that mean is $\mu$ and standard deviation is $\sigma$ and X is measure of variable.

So,       $Z=\frac{X-\mu}{\sigma}$

Given that, mean = 25 and deviation = 4

Now we have to find lower bound value of variable X when Z = -1.645

            $-1.645=\frac{X-25}{4}\\\\X=18.42$

Now we have to find upper bound value of variable X when Z = 1.645

           $1.645=\frac{X-25}{4}\\\\X=31.58$

Since, value of variable X  is between 18.42 and 31.58.

Therefore, For a simple haircut, the middle 90 percent of the customers will require between 18.4 and 31.6 minutes.

Thus, option (b) is correct.

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