Question

an elevator has a placard stating that the maximum capacity is 3600 lb—25 passengers. so, 25 adult male passengers can have a mean weight of up to 3600/25 = 144 pounds. assume that weights of males are normally distributed with a mean of 178 lb and a standard deviation of 37 lb.
a. find the probability that 1 randomly selected adult male has a weight greater than 144 lb.
b. find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 144 lb.
c. what do you conclude about the safety of this elevator?
(round to four decimal places as needed.)
c. does this elevator appear to be safe?
○ a. no, because 25 randomly selected people will never be under the weight limit.
○ b. yes, because there is a good chance that 25 randomly selected people will not exceed the elevator capacity.
○ c. yes, because 25 randomly selected adult male passengers will always be under the weight limit.
○ d. no, because there is a good chance that 25 randomly selected adult male passengers will exceed the elevator capacity.

Explanation:

Step1: Calculate z - score for part a

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 144\), \(\mu=178\), \(\sigma = 37\).

$$z=\frac{144 - 178}{37}=\frac{- 34}{37}\approx - 0.92$$

We want \(P(X>144)\), which is \(1 - P(X\leq144)\). Using the standard normal table, \(P(Z\leq - 0.92)=0.1788\). So \(P(X>144)=1 - 0.1788 = 0.8212\)

Step2: Calculate z - score for part b

The formula for the z - score of the sample mean is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}=144\), \(\mu = 178\), \(\sigma=37\), \(n = 25\)

$$z=\frac{144-178}{\frac{37}{\sqrt{25}}}=\frac{-34}{\frac{37}{5}}=\frac{-34\times5}{37}=\frac{-170}{37}\approx - 4.59$$

We want \(P(\bar{X}>144)\), which is \(1 - P(\bar{X}\leq144)\). Using the standard normal table, \(P(Z\leq - 4.59)\approx0\). So \(P(\bar{X}>144)=1-0 = 1\)

Step3: Analyze part c

Since the probability that the mean weight of 25 randomly selected adult males is greater than 144 lb (the weight limit for safety) is very high (close to 1), there is a good chance that 25 randomly selected adult male passengers will exceed the elevator capacity.

Answer:

a. \(0.8212\)
b. \(1.0000\)
c. D. No, because there is a good chance that 25 randomly selected adult male passengers will exceed the elevator capacity.