An elevator has a placard stating that the maximum capacity is 3800 lbs for 26 passengers. Thus, 26 adult male passengers can have a mean weight of up to \(\frac{3800}{26} = 146\) pounds. Assume that the weights of males are normally distributed with a mean of 182 lbs and a standard deviation of 26 lbs.

a. Find the probability that 1 randomly selected adult male has a weight greater than 146 lbs.

To calculate the probability that a randomly selected male is over 146 pounds, given a normal distribution with a mean weight of 182 pounds and a standard deviation of 26 pounds, find the Z-score for 146 pounds and use a Z-table or calculator to determine the area to the right of this score.

Explanation:

To find the probability that 1 randomly selected adult male has a weight greater than 146 pounds, given that the weights are normally distributed with a mean of 182 pounds and a standard deviation of 26 pounds, we use the standard normal distribution (Z-score).

First, calculate the Z-score for 146 pounds:

Z = (X - μ) / σ

Where X is the value of interest (146), μ is the mean (182), and σ is the standard deviation (26).

Z = (146 - 182) / 26 = -36 / 26 ≈ -1.38

Using a Z-table or calculator, we can find that the probability (P) that a randomly selected male exceeds 146 pounds is:

P(Z > -1.38)

The Z-table or normal distribution calculator will give us the area to the right of Z = -1.38, which represents our desired probability.