An elevator has a placard stating that the maximum capacity is 1296 lb long dash — 8 passengers.​ So, 8 adult male passengers can have a mean weight of up to 1296 divided by 8 equals 162 pounds. 1296/8=162 pounds. If the elevator is loaded with 8 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 162 lb.​ (Assume that weights of males are normally distributed with a mean of 170 lb and a standard deviation of 29 lb​.) Does this elevator appear to be​ safe?

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Answer:

There is a 98.26% probability that it is overloaded.

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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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 170 lb, so [tex]\mu = 170[/tex]

They have a standard deviation of standard deviation of 29 lb, so [tex]\sigma = 29[/tex].

We have a sample of 8 adults, so we have to find the standard deviation of the sample to use in the place of [tex]\sigma[/tex] in the Z score formula.

[tex]s = \frac{\sigma}{8} = \frac{29}{8} = 3.625[/tex]

Find the probability that it is overloaded because they have a mean weight greater than 162 lb, so [tex]X = 162[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{162 - 170}{3.625}[/tex]

[tex]Z = -2.21[/tex]

[tex]Z = -2.21[/tex] has a pvalue 0.0174.

So, there is a 1-0.0174 = 0.9826 = 98.26% probability that it is overloaded.

Any probability that is above 95% is considerer unusually high. So, this elevator does not appear to be safe, since there is a 98.26% probability that it is overloaded.

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