"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

For this case we want to find this probability:

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

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.

Step-by-step 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.

So we can assume

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:

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

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.