Question

Elevator Safety Example 2 referred to an elevator with a maximum capacity of 4000 lb. When rating elevators, it is common to use a 25% safety factor, so the elevator should 6-4 Basic Skills and Concepts 6-4 The Central Limit Theorem 273 actually be able to carry a load that is 25% greater than the stated limit. The maximum capacity of 4000 lb becomes 5000 lb after it is increased by 25%, so 27 adult male passengers can have a mean weight of up to 185 lb. If the elevator is loaded with 27 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 185 lb. (As in Example 2, assume that weights of males are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.) Does this elevator appear to be safe?

Answer 1

Answer:

0.7019; no, it does not appear to be safe.

Step-by-step explanation:

We want to find P(X > 185).  However, in a z table, we are given the area under the curve to the left of the value; this means we want to find

1 - P(X ≤ 185)

The z score for the mean of a sample is given by

$z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}$

For this situation, the mean, μ, is 189 and the standard deviation, σ, is 39.  Our sample size, n, is 27.  This gives us

z = (185-189)/(39÷√27) = -4/(39÷5.1962) = -4/7.5055 = -0.53

Using a z table, we see that the area under the curve to the left of this value is 0.2981.  This means our probability is

1-0.2981 = 0.7019.