Questions: An elevator has a placard stating that the maximum capacity is 3500 lb-25 passengers. So, 25 adult male passengers can have a mean weight of up to 3500 / 25=140 pounds. Assume that weights of males are normally distributed with a mean of 185 lb and a standard deviation of 40 lb. a. Find the probability that 1 randomly selected adult male has a weight greater than 140 lb. b. Find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb. c. What do you conclude about the safety of this elevator? a. The probability that 1 randomly selected adult male has a weight greater than 140 lb is 0.8697. (Round to four decimal places as needed.) b. The probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb is (Round to four decimal places as needed.)

Solution Steps

To find the probability that a randomly selected adult male weighs more than 140 lb, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} = \frac{140 - 185}{40} = -1.125 \]

Next, we find the probability that a randomly selected male weighs more than 140 lb. This is given by:

\[ P(X > 140) = 1 - P(X \leq 140) = 1 - \Phi(-1.125) \approx 0.1303 \]

Thus, the probability that 1 randomly selected male weighs more than 140 lb is approximately \(0.1303\).

Now, we calculate the probability that the mean weight of a sample of 25 randomly selected adult males is greater than 140 lb. The standard error of the mean is given by:

\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{40}{\sqrt{25}} = 8 \]

We then calculate the Z-score for the sample mean:

\[ z = \frac{140 - 185}{8} = -5.625 \]

The probability is then calculated as:

\[ P(\bar{X} > 140) = 1 - P(\bar{X} \leq 140) = 1 - \Phi(-5.625) \approx 1.0 \]

Thus, the probability that a sample of 25 males has a mean weight greater than 140 lb is approximately \(1.0\).

Given that the probability that a sample of 25 adult males has a mean weight greater than 140 lb is \(1.0\), we conclude that the elevator is likely safe for 25 adult males.

Final Answer