Questions: An airliner carries 100 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in. Complete parts (a) through (d). a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending. The probability is (Round to four decimal places as needed.)

An airliner carries 100 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in. Complete parts (a) through (d).
a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

The probability is
(Round to four decimal places as needed.)

Transcript text: An airliner carries 100 passengers and has doors with a height of 72 in . Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in . Complete parts (a) through (d). a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending. The probability is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Problem

We need to find the probability that a randomly selected male passenger can fit through a doorway with a height of $72$ inches. The heights of men are normally distributed with a mean ($\mu$) of $69.0$ inches and a standard deviation ($\sigma$) of $2.8$ inches.

Step 2: Set Up the Normal Distribution

The probability we are looking for can be expressed as: \[ P(X \leq 72) = \Phi\left(\frac{72 - \mu}{\sigma}\right) \] where $X$ is the height of a randomly selected male passenger.

Step 3: Calculate the Z-Score

We calculate the Z-score for the upper bound (door height): \[ Z_{end} = \frac{72 - 69.0}{2.8} = \frac{3.0}{2.8} \approx 1.0714 \] The Z-score for the lower bound is: \[ Z_{start} = -\infty \]

Step 4: Calculate the Probability

Using the Z-scores, we find the probability: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0714) - \Phi(-\infty) \] Since $\Phi(-\infty) = 0$, we have: \[ P = \Phi(1.0714) \approx 0.858 \]