A luxury passenger liner has 500 passengers on board whose ages are normally distributed around a mean of 60 years with a standard deviation of 12 years. How many passengers are between 45 and 78 years old?

Answer 1

To solve this, we need to find the percentage of passengers whose ages are between 45 and 78 years old.

First, we'll standardize the lower and upper limits of the age range using the z-score formula:

For the lower limit of 45 years old:
z1 = (45 - 60) / 12 = -1.25

For the upper limit of 78 years old:
z2 = (78 - 60) / 12 = 1.5

Next, we'll find the corresponding percentage for each z-score using a standard normal distribution table or a calculator.

For the lower limit:
P(Z < -1.25) ≈ 0.1056 (approximately)

For the upper limit:
P(Z < 1.5) ≈ 0.9332 (approximately)

To find the percentage of passengers between the two limits, we subtract the lower limit percentage from the upper limit percentage:
P(-1.25 < Z < 1.5) = 0.9332 - 0.1056 = 0.8276

Finally, we multiply the percentage by the total number of passengers:
0.8276 * 500 ≈ 413.8

Approximately 414 passengers are between 45 and 78 years old.