A First Course in Probability (10th Edition)
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Transcribed Image Text:### Elevator Capacity and Passenger Weight Analysis
An elevator has a placard stating that the maximum capacity is 4100 lb—27 passengers. Therefore, 27 adult male passengers can have a mean weight of up to \( \frac{4100}{27} = 152 \) pounds. Assume that weights of males are normally distributed with a mean of 185 lb and a standard deviation of 29 lb.
#### Questions:
**a.** Find the probability that 1 randomly selected adult male has a weight greater than 152 lb.
**b.** Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 152 lb.
**c.** What do you conclude about the safety of this elevator?
### Explanation:
To solve this problem, the concept of the normal distribution and statistical sampling will be applied.
1. **For part (a):** Use the normal distribution with a mean of 185 lb and a standard deviation of 29 lb to find the probability that an individual weighs more than 152 lb.
2. **For part (b):** Use the sampling distribution of the sample mean. The mean remains the same (185 lb), but the standard deviation (the standard error) will be the population standard deviation divided by the square root of the sample size (\( \frac{29}{\sqrt{27}} \)). Find the probability that the sample mean weight is greater than 152 lb.
3. **For part (c):** Analyze the probabilities calculated in parts (a) and (b) to assess whether the elevator's capacity guidelines are likely to be safe, given the distribution of weights.
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