Question

an elevator has a placard stating that the maximum capacity is 3700 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 190 lb and a standard deviation of 32 lb.
a. find the probability that 1 randomly selected adult male has a weight greater than 142 lb.
b. find the probability that a sample of 26 randomly selected adult males has a mean weight greater than 142 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 142 lb is (round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score for a single adult male

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 142\), \(\mu=190\), and \(\sigma = 32\).

$$z=\frac{142 - 190}{32}=\frac{-48}{32}=-1.5$$

Step2: Find the probability using the standard normal distribution

We want \(P(X>142)\). Since \(P(X > x)=1 - P(X\leq x)\), and for \(z=-1.5\), from the standard normal table \(P(Z\leq - 1.5)=0.0668\)

$$P(X>142)=1 - 0.0668 = 0.9332$$

Answer:

\(0.9332\)