Question

An elevator has a placard stating that the maximum capacity is 19201920 lblong dash—1212 passengers.​ So, 1212 adult male passengers can have a mean weight of up to 1920 divided by 12 equals 160 pounds.1920/12=160 pounds. If the elevator is loaded with 1212 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 160160 lb.​ (Assume that weights of males are normally distributed with a mean of 163 lb163 lb and a standard deviation of 29 lb29 lb​.) Does this elevator appear to be​ safe?

Answer 1

Answer:

This means that there is a 1-0.3594 = 0.6406 = 64.06% probability that the elevator is overloaded. This a good chance that the elevator's limit weight will be exceeded. So, this elevator does not appear to be safe.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by

$Z = \frac{X - \mu}{\sigma}$

After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X. Subtracting 1 by the pvalue, we This p-value is the probability that the value of the measure is greater than X.

In this problem, we have that:

Assume that weights of males are normally distributed with a mean of 163 lb and a standard deviation of 29 lb.

This means that $\mu = 163, \sigma = 29$.

We have a sample of 12 adults, and we want to calculate the zscore of THE SAMPLE'S AVERAGE so we need to find the standard deviation of the sample. This is

$s = \frac{\sigma}{\sqrt{12}} = 8.37$

Find the probability that it is overloaded because they have a mean weight greater than 160.

This is 1 subtracted by the pvalue of Z when $X = 160$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{160 - 163}{8.37}$

$Z = -0.36$

$Z = -0.36$ has a pvalue of 0.3594.

This means that there is a 1-0.3594 = 0.6406 = 64.06% probability that the elevator is overloaded. This a good chance that the elevator's limit weight will be exceeded. So, this elevator does not appear to be safe.