An elevator has a placard stating that the maximum capacity is $4400 \mathrm{lb}-29$ passengers. So, 29 adult male passengers can have a mean weight of up to $4400 / 29=152$ pounds. Assume that weights of males are normally distributed with a mean of $188 \mathrm{lb}$ and a standard deviation of $37 \mathrm{lb}$. a. Find the probability that 1 randomly selected adult male has a weight greater than $152 \mathrm{lb}$. b. Find the probability that a sample of 29 randomly selected adult males has a mean weight greater than $152 \mathrm{lb}$. 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 $152 \mathrm{lb}$ is (Round to four decimal places as needed.)

Step 1 ：We are given that the mean weight of adult males is 188 pounds and the standard deviation is 37 pounds. We are asked to find the probability that a randomly selected adult male has a weight greater than 152 pounds.

Step 2 ：To solve this, we first calculate the Z-score for a weight of 152 pounds. The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 3 ：Substituting the given values into the formula, we get \(Z = \frac{152 - 188}{37} = -0.972972972972973\).

Step 4 ：The Z-score tells us how many standard deviations below the mean a weight of 152 pounds is. A negative Z-score indicates that the value is below the mean.

Step 5 ：We then find the probability that a randomly selected adult male has a weight greater than this Z-score. This is equivalent to finding the area to the right of the Z-score on the standard normal distribution curve.

Step 6 ：Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.8347166325829531.

Step 7 ：Rounding to four decimal places, the probability that a randomly selected adult male has a weight greater than 152 pounds is approximately 0.8347. This means that about 83.47% of adult males weigh more than 152 pounds.

Step 8 ：Final Answer: The probability that 1 randomly selected adult male has a weight greater than 152 pounds is approximately \(\boxed{0.8347}\).