Questions: An elevator has a placard stating that the maximum capacity is 4500 lb-29 passengers. So, 29 adult male passengers can have a mean weight of up to 4500 / 29=155 pounds. Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of 33 lb. a. Find the probability that 1 randomly selected adult male has a weight greater than 155 lb. b. Find the probability that a sample of 29 randomly selected adult males has a mean weight greater than 155 lb. c. What do you conclude about the safety of this elevator?

An elevator has a placard stating that the maximum capacity is 4500 lb-29 passengers. So, 29 adult male passengers can have a mean weight of up to 4500 / 29=155 pounds. Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of 33 lb. a. Find the probability that 1 randomly selected adult male has a weight greater than 155 lb. b. Find the probability that a sample of 29 randomly selected adult males has a mean weight greater than 155 lb. c. What do you conclude about the safety of this elevator?
Transcript text: An elevator has a placard stating that the maximum capacity is $4500 \mathrm{lb}-29$ passengers. So, 29 adult male passengers can have a mean weight of up to $4500 / 29=155$ pounds. Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of 33 lb . a. Find the probability that 1 randomly selected adult male has a weight greater than 155 lb . b. Find the probability that a sample of 29 randomly selected adult males has a mean weight greater than 155 lb . c. What do you conclude about the safety of this elevator?
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Solution

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Solution Steps

Step 1: Calculating the probability for 1 randomly selected adult male

Given a maximum capacity of 4500lb for 29 passengers, each passenger's mean weight limit is 155.1724lb. Using the Z-score formula for a single observation: $Z = \frac{X - \mu}{\sigma} = \frac{155.172 - 1}{33} = -0.752.$ The probability that 1 randomly selected adult male has a weight greater than 155.1724lb is approximately 0.774.

Step 2: Calculating the probability for a sample of N randomly selected adult males

Using the Z-score formula for a sample mean: $Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{N}} = \frac{155.172 - 1}{33 / \sqrt{29}} = -4.051.$ The probability that a sample of 29 randomly selected adult males has a mean weight greater than 155.1724lb is approximately 1.

Final Answer:

Based on the calculated probabilities, the elevator may not be considered safe due to a high probability of overload.

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