Answer: The lavatory door should be at least 212.2738 cm (or about 7 feet) high to ensure that 99.999% of adult males will not have to stop as they enter.
Explanation: To ensure that 99.999% of adult males will not have to stop as they enter the aircraft lavatory, we need to determine the height of the lavatory door such that only the tallest 0.001% of adult males will have to stop.
To find this height, we need to use the inverse normal cumulative distribution function, which gives the z-score corresponding to a given percentile. In this case, we want to find the z-score corresponding to the 99.999th percentile, which is given by:
z = invNorm(0.99999) ≈ 4.7534
where invNorm is the inverse normal cumulative distribution function.
We can then use the z-score formula to find the corresponding height:
z = (x - μ) / σ
where x is the height we want to find, μ is the mean height (180 cm), and σ is the standard deviation (7 cm). Solving for x, we get:
x = z * σ + μ
= 4.7534 * 7 + 180
≈ 212.2738 cm