Question

an elevator has a placard stating that the maximum capacity is 4000 lb—27 passengers. so, 27 adult male passengers can have a mean weight of up to 4000/27 = 148 pounds. assume that weights of males are normally distributed with a mean of 188 lb and a standard deviation of 26 lb.
a. find the probability that 1 randomly selected adult male has a weight greater than 148 lb.
b. find the probability that a sample of 27 randomly - selected adult males has a mean weight greater than 148 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 148 lb is
(round to four decimal places as needed.)

Explanation:

Step1: Calculate z - score for single - male case

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 148$, $\mu=188$, and $\sigma = 26$.
$z=\frac{148 - 188}{26}=\frac{- 40}{26}\approx - 1.54$

Step2: Find the probability for single - male case

We want $P(X>148)$. Since the total area under the normal curve is 1, and $P(X>148)=1 - P(X\leq148)$. Looking up the z - score of - 1.54 in the standard normal table, $P(Z\leq - 1.54)=0.0618$. So $P(X>148)=1 - 0.0618 = 0.9382$.

Step3: Calculate z - score for sample - mean case

The standard deviation of the sample mean (also known as the standard error) is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 26$, $n = 27$. So $\sigma_{\bar{x}}=\frac{26}{\sqrt{27}}\approx4.98$. The z - score formula for the sample mean is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, with $\bar{x}=148$, $\mu = 188$. Then $z=\frac{148 - 188}{4.98}=\frac{-40}{4.98}\approx - 8.03$.

Step4: Find the probability for sample - mean case

We want $P(\bar{X}>148)$. Since $P(\bar{X}>148)=1 - P(\bar{X}\leq148)$. Looking up the z - score of - 8.03 in the standard normal table (values beyond - 3.49 have a very small probability), $P(Z\leq - 8.03)\approx0$. So $P(\bar{X}>148)=1-0 = 1$.

Step5: Conclusion about elevator safety

The probability that a single male's weight exceeds 148 lb is relatively high (0.9382). And the probability that the mean weight of 27 male passengers exceeds 148 lb is almost 1. This indicates that the elevator is likely to be over - loaded when carrying 27 adult male passengers, so it is not very safe under these assumptions.

Answer:

a. 0.9382
b. 1
c. The elevator is likely to be over - loaded when carrying 27 adult male passengers, so it is not very safe under these assumptions.