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 180 lb and a standard deviation of 26 lb.
a. find the probability that 1 randomly selected adult male has a weight greater than 142 lb.
(round to four decimal places as needed.)
.9279
...
b. find the probability that a sample of 26 randomly selected adult males has a mean weight greater than 142 lb.
(round to four decimal places as needed.)
1
c. does this elevator appear to be safe?
a. yes, because there is a good chance that 26 randomly selected people will not exceed the elevator capacity.
b. no, because 26 randomly selected people will never be under the weight limit.
c. yes, because 26 randomly selected adult male passengers will always be under the weight limit.
d. no, because there is a good chance that 26 randomly selected adult male passengers will exceed the elevator capacity.

Explanation:

Step1: Calculate the standard error for part b

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 26$ and $n = 26$, we have $\sigma_{\bar{x}}=\frac{26}{\sqrt{26}}\approx5.0990$.

Step2: Calculate the z - score for part b

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x}=142$, $\mu = 180$, and $\sigma_{\bar{x}}\approx5.0990$. So $z=\frac{142 - 180}{5.0990}\approx - 7.45$.

Step3: Find the probability for part b

Using the standard normal distribution table (or a calculator with normalcdf function, where for $z=-7.45$, the probability $P(\bar{X}>142)=1 - P(\bar{X}\leq142)$. Since for a standard normal variable $Z$, when $z=-7.45$, $P(Z\leq - 7.45)\approx0$. So $P(\bar{X}>142)\approx1$.

Step4: Analyze part c

The maximum capacity is $3700$ lb for $26$ passengers, with a mean weight limit of $\frac{3700}{26}=142$ lb. The probability that a sample of $26$ adult males has a mean weight greater than $142$ lb is approximately $1$ (from part b). This means there is a very high chance that the total weight of $26$ randomly - selected adult males will exceed the elevator capacity.

Answer:

b. $1$; c. D. No, because there is a good chance that 26 randomly selected adult male passengers will exceed the elevator capacity.