Question

At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 22 minutes and a standard deviation of 3 minutes. If you visit that restaurant 37 times this year, what is the expected number of times that you would expect to wait between 19 minutes and 23 minutes, to the nearest whole number?

Answer 1

To find the expected number of times you would wait between 19 and 23 minutes, we need to calculate the z-scores for these values and use them to find the area under the normal distribution curve between those values.

First, we calculate the z-score for 19 minutes:

z = (19 - 22) / 3 = -1

Next, we calculate the z-score for 23 minutes:

z = (23 - 22) / 3 = 0.33

Using a standard normal distribution table or calculator, we can find the area under the normal distribution curve between these z-scores:

P(-1 < z < 0.33) = 0.4082 - 0.3413 = 0.0669

This means that there is a probability of 0.0669 of waiting between 19 and 23 minutes for a single visit to the restaurant.

To find the expected number of times you would wait between 19 and 23 minutes over 37 visits, we multiply the probability for a single visit by the number of visits:

Expected number of times = 0.0669 x 37 ≈ 2.47

Rounding to the nearest whole number, we would expect to wait between 19 and 23 minutes about 2 times over 37 visits to the restaurant.