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.
2.1 How many of the passengers are between 45 and 78 years old?
2.2 What percentage of the passengers is between 48 and 72 years old?
2.1 To find out how many passengers are between 45 and 78 years old, we need to find the z-scores for both ages and then calculate the proportion of passengers between those z-scores in a standard normal distribution.
For 45 years:
z = (45 - 60) / 12 = -1.25
For 78 years:
z = (78 - 60) / 12 = 1.50
Using a standard normal distribution table or calculator, we find:
- The proportion to the left of z = -1.25 is 0.1056
- The proportion to the left of z = 1.50 is 0.9332
Therefore, the proportion of passengers between 45 and 78 years old is 0.9332 - 0.1056 = 0.8276.
Since there are 500 passengers in total, the number of passengers between 45 and 78 years old would be:
0.8276 * 500 = 413.8
So, approximately 414 passengers are between 45 and 78 years old.
2.2 To find the percentage of passengers between 48 and 72 years old, we repeat the same process.
For 48 years:
z = (48 - 60) / 12 = -1.00
For 72 years:
z = (72 - 60) / 12 = 1.00
Using a standard normal distribution table or calculator, we find:
- The proportion to the left of z = -1.00 is 0.1587
- The proportion to the left of z = 1.00 is 0.8413
Therefore, the percentage of passengers between 48 and 72 years old is (0.8413 - 0.1587) * 100 = 68.26%.
So, approximately 68.26% of the passengers are between 48 and 72 years old.