Questions: 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 5. Using the empirical rule (as presented in the book), what is the approximate percentage of daily phone calls numbering between 32 and 42? ans = %

Solution Steps

To find the probability of the number of phone calls answered between 32 and 42, we first calculate the Z-scores for the lower and upper bounds using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

For the lower bound \(X = 32\):

\[ Z_{start} = \frac{32 - 37}{5} = -1.0 \]

For the upper bound \(X = 42\):

\[ Z_{end} = \frac{42 - 37}{5} = 1.0 \]

Next, we use the standard normal distribution to find the probabilities corresponding to these Z-scores. The probability that the number of phone calls is between 32 and 42 is given by:

\[ P(32 < X < 42) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) \]

Using standard normal distribution tables or functions, we find:

\[ \Phi(1.0) \approx 0.8413 \quad \text{and} \quad \Phi(-1.0) \approx 0.1587 \]

Thus,

\[ P(32 < X < 42) = 0.8413 - 0.1587 = 0.6826 \]

To express this probability as a percentage, we multiply by 100:

\[ P(32 < X < 42) \times 100 \approx 68.26\% \]

Final Answer