An elevator has a placard stating that the maximum capacity is $3700 \mathrm{lb}-26$ passengers. So, 26 adult male passengers can have a mean weight of up to $3700 / 26=142$ pounds. Assume that weights of males are normally distributed with a mean of $187 \mathrm{lb}$ and a standard deviation of $39 \mathrm{lb}$.
a. The probability that 1 randomly selected adult male has a weight greater than $142 \mathrm{lb}$ is (Round to four decimal places as needed.)
b. The probability that a sample of 26 randomly selected adult males has a mean weight gregater than $142 \mathrm{lb}$ is (Round to four decimal places as needed.)
c. Does this elevator appear to be safe?
A. No, because there is a good chance that 26 randomly selected adult male passengers will exceed the elevator capacity.
B. Yes, because there is a good chance that 26 randomly selected people will not exceed the elevator capacity.
C. Yes, because 26 randomly selected adult male passengers will always be under the weight limit.
D. No, because 26 randomly selected people will never be under the weight limit.
Answer
\(\boxed{0.8757}\) is the probability that 1 randomly selected adult male has a weight greater than 142 pounds.
Step 1 ：We are given that the weights of males are normally distributed with a mean of 187 pounds and a standard deviation of 39 pounds.
Step 2 ：We are asked to find the probability that 1 randomly selected adult male has a weight greater than 142 pounds.
Step 3 ：We can use the z-score formula to find the z-score for a weight of 142 pounds. The z-score formula is \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 4 ：Substituting the given values into the z-score formula, we get \(Z = \frac{142 - 187}{39} = -1.1538461538461537\).
Step 5 ：We then use a z-table or a statistical function to find the probability that a randomly selected male has a weight greater than 142 pounds. The probability corresponding to a z-score of -1.1538461538461537 is approximately 0.8757.
Step 6 ：\(\boxed{0.8757}\) is the probability that 1 randomly selected adult male has a weight greater than 142 pounds.
