An airliner carries 350 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:

B. If half the passengers are men, find the probability that the mean height of the 175 men is less than 72 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: A. To find the probability that a randomly selected male passenger can fit through the doorway without bending, we need to find the probability that his height is less than or equal to 72 inches. Since the heights of men are normally distributed with a mean of 69 inches and a standard deviation of 2.8 inches, we can use the standard normal distribution to find this probability.The standard normal distribution has a mean of 0 and a standard deviation of 1. We can convert the height of a male passenger from the given distribution to the standard normal distribution by using the formula:$z = \frac{x – \mu}{\sigma}$where $x$ is the height of the male passenger, $\mu$ is the mean height of 69 inches, and $\sigma$ is the standard deviation of 2.8 inches. Substituting these values into the formula gives us:$z = \frac{72 – 69}{2.8} = \frac{3}{2.8} = 1.0714$Since the standard normal distribution is symmetric, the probability that a male passenger’s height is less than or equal to 72 inches is the same as the probability that his height is greater than or equal to 72 inches. We can find this probability by looking up the value of $z = 1.0714$ in a standard normal table, which gives us a probability of 0.8413. Therefore, the probability that a male passenger can fit through the doorway without bending is 0.8413.B. To find the probability that the mean height of the 175 men is less than 72 inches, we need to find the probability that the sample mean is less than 72 inches. Since the heights of men are normally distributed with a mean of 69 inches and a standard deviation of 2.8 inches, the sample mean will also be normally distributed with a mean of 69 inches and a standard deviation of $\frac{2.8}{\sqrt{175}} = 0.32$. We can use the standard normal distribution to find the probability that the sample mean is less than 72 inches.The standard normal distribution has a mean of 0 and a standard deviation of 1. We can convert the sample mean from the given distribution to the standard normal distribution by using the formula:$z = \frac{x – \mu}{\sigma}$where $x$ is the sample mean, $\mu$ is the mean of 69 inches, and $\sigma$ is the standard deviation of 0.32. Substituting these values into the formula gives us:$z = \frac{72 – 69}{0.32} = \frac{3}{0.32} = 9.375$Since the standard normal distribution is symmetric, the probability that the sample mean is less than 72 inches is the same as the probability that the sample mean is greater than 72 inches. We can find this probability by looking up the value of $z = 9.375$ in a standard normal table, which gives us a probability of 0.0000. Therefore, the probability that the mean height of the 175 men is less than 72 inches is 0.0000.C. When considering the comfort and safety of passengers, the result from part (a) is more relevant than the result from part (b). This is because part (a) tells us the probability that a randomly selected male passenger can fit through the doorway without bending, which is directly relevant to the comfort and safety of the passengers. In contrast, part (b) tells us the probability that the mean height of the 175 men is less than 72 inches, which is not directly relevant to the comfortD. Women are ignored in this case because the problem only asks about the heights of men. The problem states that the heights of men are normally distributed with a mean of 69 inches and a standard deviation of 2.8 inches, and does not provide any information about the heights of women. Therefore, we cannot use the given information to find the probability that a female passenger can fit through the doorway without bending.