Question

A) If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending. The probability is: B) If half of the passengers are men, find the probability that the mean height of the men is less than 74 in. 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? D) When considering the comfort and safety of passengers, why are women ignored in this case?

Answer 1

Hello there. To solve this question, we'll have to remember some properties about probabilities.

Selecting male passengers randomly from the 400 passengers set, we want to find the probability that he can fit through the 74 in doorway without bending.

First, calling x the height of the random selected man, the probability of him fitting is P(x <= 74), and of him not fitting is P(x > 74)

To find P(x <= 74), we calculate the z-score by the formula:

`z=(x-\mu)/(\sigma)=(74-69)/(2.8)=(5)/(2.8)\approx1.78`

Looking for a table of z-scores for a normally distributed set, we say that the probability is around 95 or 96%. Using the under 8 approximation, P(x <= 74) = 0.96246 or 96.25%.

part (b)

We now have that the sample size is 200, since we had 400 passengers on the flight.

Making the new standard deviation as:

`s=\frac{\sigma}{\sqrt[]{\text{sample}}}=\frac{2.8}{\sqrt[]{200}}=0.198`

From the Central limit theorem, the z-score will then be:

z = (74 - 69)/0.198 = 25.25