Question

"In a mid-size company, the distribution of the number of phone calls answered each day by each of the 12 receptionists is bell-shaped and has a mean of 37 and a standard deviation of 9. Using the Empirical Rule (as presented in the book), what is the approximate percentage of daily phone calls numbering between 10 and 64

Answer 1

For this case we want to find this probability:

`P(10<X<64)`

And we can use the z score formula to see how many deviation we are within the mean and we got:

`z = (10-37)/(9)=-3`

`z = (64-37)/(9)=3`

And for this case we know that within 3 deviation from the mean we have 99.7% of the values and that's the answer for this case.

Explanation:

Previous concepts

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the number of phone calls answered.

From the problem we have the mean and the standard deviation for the random variable X.
`E(X)=37, Sd(X)=9`

So we can assume
`\mu=37 , \sigma=9`

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

Solution to the problem

For this case we want to find this probability:

`P(10<X<64)`

And we can use the z score formula to see how many deviation we are within the mean and we got:

`z = (10-37)/(9)=-3`

`z = (64-37)/(9)=3`

And for this case we know that within 3 deviation from the mean we have 99.7% of the values and that's the answer for this case.