Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B).


a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

Verified step by step guidance

1. Step 1: Identify the key components of the problem. The doorway height is 72 inches, the mean height of men is 68.6 inches, and the standard deviation is 2.8 inches. Heights are normally distributed, so we will use the standard normal distribution (Z-distribution) to solve this problem.
2. Step 2: Define the random variable X as the height of a randomly selected male passenger. To find the probability that a male passenger can fit through the doorway without bending, we need to calculate P(X ≤ 72).
3. Step 3: Standardize the random variable X to convert it into a Z-score using the formula: Z = (X - μ) / σ, where X is the value of interest (72 inches), μ is the mean (68.6 inches), and σ is the standard deviation (2.8 inches).
4. Step 4: Substitute the values into the Z-score formula: Z = (72 - 68.6) / 2.8. Simplify the numerator and divide by the standard deviation to compute the Z-score.
5. Step 5: Use a standard normal distribution table or a statistical software to find the cumulative probability corresponding to the calculated Z-score. This cumulative probability represents the probability that a randomly selected male passenger can fit through the doorway without bending.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. In this context, the heights of men are normally distributed, which allows us to use the properties of the normal curve to calculate probabilities related to height.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this problem, the Z-score will help determine how many standard deviations a male passenger's height is from the mean, which is essential for finding the probability of fitting through the doorway.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring. In this scenario, we need to calculate the probability that a randomly selected male passenger's height is less than or equal to the doorway height of 72 inches. This is done using the Z-score and standard normal distribution tables or software to find the corresponding probability.