We are tasked with finding the probability that a randomly selected male passenger can fit through the doorway of an airliner without bending. This can be modeled using a normal distribution.

Given dataWe are tasked with finding the probability that a randomly selected male passenger can fit through the doorway of an airliner without bending. This can be modeled using a normal distribution.

Given

• Mean height of men: $\mu = 69.0 \text{ inches}$
• Standard deviation of heights: $\sigma = 2.8 \text{ inches}$
• Height of the door: $78 \text{ inches}$

The problem is asking for the probability that the height of a randomly selected male passenger is less than or equal to $78$ inches, which corresponds to finding the cumulative probability in a normal distribution.

Step 1: Standardize the height

We can standardize the value of 78 inches using the Z-score formula: $Z = \frac{X - \mu}{\sigma}$ where:

• $X = 78 \text{ inches}$ (the height of the door),
• $\mu = 69.0 \text{ inches}$ (mean height of men),
• $\sigma = 2.8 \text{ inches}$ (standard deviation).

Substituting the values: $Z = \frac{78 - 69.0}{2.8} = \frac{9}{2.8} \approx 3.21$

Step 2: Find the cumulative probability

Using the Z-score table (or a normal distribution calculator), a Z-score of $3.21$ corresponds to a cumulative probability of approximately 0.9993.

Thus, the probability that a randomly selected male passenger can fit through the doorway without bending is:

$\boxed{0.9993}$

This means there is a 99.93% chance that a male passenger can fit through the door without bending.

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Would you like to explore more details or have any other questions?
Here are 5 related questions to deepen your understanding:

1. How do we compute Z-scores for values below the mean?
2. What is the significance of the Z-score in the context of normal distribution?
3. How would the result change if the doorway height were lower, say 75 inches?
4. Can we apply the same method for female passengers if given their mean height and standard deviation?
5. How does increasing the standard deviation affect the probability in a normal distribution?

Tip: Always use the Z-score formula to standardize values before looking them up in a normal distribution table.