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


Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000/27=148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.


c. What do you conclude about the safety of this elevator?

Verified step by step guidance

1. Step 1: Identify the problem. The elevator has a maximum capacity of 4000 lb for 27 passengers, which implies an average weight of 148 lb per passenger. We need to assess whether the elevator is safe when filled with 27 adult males whose weights are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.
2. Step 2: Define the random variable. Let X represent the weight of a single adult male. Since the weights are normally distributed, the total weight of 27 adult males can be modeled as the sum of 27 independent random variables, each with mean 189 lb and standard deviation 39 lb.
3. Step 3: Calculate the mean and standard deviation of the total weight. The mean of the total weight is given by \( \mu_{total} = n \cdot \mu \), where \( n = 27 \) and \( \mu = 189 \). The standard deviation of the total weight is given by \( \sigma_{total} = \sqrt{n} \cdot \sigma \), where \( \sigma = 39 \).
4. Step 4: Determine the probability distribution of the total weight. Since the individual weights are normally distributed, the sum of 27 weights will also follow a normal distribution with the calculated mean and standard deviation. Use this distribution to find the probability that the total weight exceeds the elevator's maximum capacity of 4000 lb.
5. Step 5: Use the z-score formula to calculate the probability. The z-score is given by \( z = \frac{X - \mu_{total}}{\sigma_{total}} \), where \( X = 4000 \). Once the z-score is calculated, use the standard normal distribution table or software to find the corresponding probability. If the probability of exceeding 4000 lb is high, the elevator may not be safe under the given assumptions.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this context, the weights of adult males are assumed to follow a normal distribution with a specified mean and standard deviation, which allows for the calculation of probabilities related to weight and safety.

Mean and Standard Deviation

The mean is the average value of a data set, while the standard deviation measures the amount of variation or dispersion from the mean. In this scenario, the mean weight of adult males is 189 lb, and the standard deviation is 39 lb, which helps assess how likely it is for a group of 27 adult males to exceed the elevator's weight limit when their weights are normally distributed.

Safety Assessment

Safety assessment involves evaluating whether a system or structure can operate within its specified limits without risk of failure. In this case, it requires analyzing the probability that the combined weight of 27 adult males exceeds the elevator's maximum capacity of 4000 lb, using statistical methods to determine if the elevator can safely accommodate the expected load.