Final answer:
Using the normal distribution, the probability that a randomly selected male passenger can fit through a door with a height of 72 inches without bending, given the men have an average height of 69.0 inches with a standard deviation of 2.8 inches, is approximately 85.77%.
Step-by-step explanation:
To determine the probability that a male passenger can fit through the door without bending, we utilize the normal distribution properties. Given the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches, and the door height is 72 inches, we want to find the probability of a man being shorter than 72 inches.
Calculating the Probability
To do so, we calculate the Z-score using the formula:
Z = (X - μ) / σ
Where X is 72 inches (door height), μ is the mean height of men (69.0 inches), and σ is the standard deviation (2.8 inches). Plugging these values into the formula gives us:
Z = (72 - 69.0) / 2.8
Z = 3 / 2.8
Z ≈ 1.0714
Now, we look up the Z-score in a standard normal distribution table, or use a calculator to find the corresponding probability. A Z-score of 1.0714 corresponds to a probability of approximately 0.8577. This means there is an 85.77% chance that a randomly selected male passenger can fit through the door without bending.