Question

At a customer service call center for a large company, the number of calls received per hour is normally distributed with a mean of 130 calls and a standard deviation of 5 calls. What is the probability that during a given hour of the day there will be between 117 calls and 126 calls, to the nearest thousandth?

Answer 1

To find the probability that there will be between 117 and 126 calls in a given hour at the customer service call center, calculate the z-scores for both values using the formula z = (X - μ) / σ and then use the standard normal distribution table. Subtract the probability associated with the larger z-score from the probability associated with the smaller z-score to find the probability that the number of calls falls between 117 and 126.

Step-by-step explanation:

To find the probability that there will be between 117 and 126 calls in a given hour, we need to calculate the z-scores for both values and use the standard normal distribution table.

First, we calculate the z-score for 117 calls:

z = (X - μ) / σ

where X is the value we're interested in, μ is the mean, and σ is the standard deviation.

z = (117 - 130) / 5 = -2.6

Next, we calculate the z-score for 126 calls:

z = (126 - 130) / 5 = -0.8

Using the standard normal distribution table, we find the probabilities associated with these z-scores:

P(z > -2.6) ≈ 0.995

P(z > -0.8) ≈ 0.788

To find the probability that the number of calls falls between 117 and 126, we subtract the probability associated with -2.6 from the probability associated with -0.8:

P(117 ≤ X ≤ 126) = P(z > -0.8) - P(z > -2.6) ≈ 0.788 - 0.995 ≈ 0.207

Therefore, the probability that there will be between 117 and 126 calls in a given hour is approximately 0.207.