Questions: A survey found that women's heights are normally distributed with mean 62.7 in. and standard deviation 3.7 in. The survey also found that men's heights are normally distributed with mean 69.7 in. and standard deviation 3.1 in. Consider an executive jet that seats six with a doorway height of 56.4 in. Complete parts (a) through (c) below.

Solution Steps

To determine if individuals can fit through the doorway of height $56.4$ in, we first calculate the Z-scores for both women's and men's heights.

For women: $$z = \frac{X - \mu}{\sigma} = \frac{56.4 - 62.7}{3.7} = -1.7027$$

For men: $$z = \frac{X - \mu}{\sigma} = \frac{56.4 - 69.7}{3.1} = -4.2903$$

Next, we calculate the probabilities that a randomly selected woman and man can fit through the door without bending.

For women: $$P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.7027) - \Phi(-\infty) = 0.0443$$

For men: $$P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-4.2903) - \Phi(-\infty) = 0.0$$

The probabilities indicate that:

• The probability that a woman can fit without bending is $0.0443$.
• The probability that a man can fit without bending is $0.0$.

Since the probability for women is quite low and for men is zero, it suggests that the door design is inadequate for both genders.

Final Answer