Question

An airliner carries 100 passengers and has doors with a height of 72 inches. Heights of men are normally distributed with a mean of 69 inches and a standard deviation of 3 inches.

a) If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.
b) If half of the 100 passengers are men, find the probability that the mean height of the 50 men is less than 72 inches.
c) When considering the comfort and safety of passengers, which result is more relevant: the probability from part (a) or the probability from part (b)? Why?

Answer 1

The given problem involves calculating probabilities using the normal distribution for the heights of male passengers in relation to the height of an aircraft's doors, and deciding which probability is more relevant for engineers designing aircraft doors.

Step-by-step explanation:

The question involves applying principles of the normal distribution to context of airline door heights and passenger heights. Specifically, it looks at the probability of a man being able to fit through an airline door without bending, and the probability distribution of male height in a group, as well as considerations for airline engineers in door design.

Part A

Given the normally distributed heights of men with a mean of 69 inches and a standard deviation of 3 inches, we can calculate the probability that a randomly selected male passenger can fit through a 72-inch door without bending using the Z-score formula: Z = (X - Mean) / Standard Deviation. Here X is 72 inches (the door height), Mean is 69 inches, and Standard Deviation is 3 inches. After calculating the Z-score, we'd look up this value in a standard normal distribution table or use software to calculate the corresponding probability.

Part B

For the second question, we use the Central Limit Theorem to determine that the sampling distribution of the sample mean for 50 men will also be normally distributed. The probability can be found by standardizing the sample mean using the standard error, which is the standard deviation divided by the square root of the sample size (n), and then calculating the cumulative probability for this standardized mean being less than 72 inches.

Part C

From a comfort and safety perspective, the result from Part B is more relevant, as it takes the average height into consideration for all male passengers rather than the height of an individual. This consideration is essential in engineering to ensure the design accommodates most passengers.