Question

Part 1

An elevator has a placard stating that the maximum capacity is 1376 lb—8 passengers.​ So, 8 adult male passengers can have a mean weight of up to 1376/8=172 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 172 lb.​ (Assume that weights of males are normally distributed with a mean of 178 lb and a standard deviation of 27 lb​.) Does this elevator appear to be​ safe?

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Part 1

The probability the elevator is overloaded is

.

Answer 1

Given : Mean $(\mu)=174$ , standard deviation $(\sigma)=28$ , sample size $(n)=8$

Formula: $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$

Find: $P\text{( sample mean greater than 172)}=P(\bar{x}>172)=?$

$\begin{aligned} &\Rightarrow P\left(\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}>\frac{172-174}{\frac{28}{\sqrt{8}}}\right) \\ &\Rightarrow P(z>-0.2020) \end{aligned}$

Note: $P(z>a)=1-P(z<a)$

$\Rightarrow 1-P(z<-0.2020)$

Refer Standard normal table/Z-table to find the probability or use excel formula "=NORM.S.DIST(-0.2020, TRUE)" to find the probability.

$\begin{aligned} &\Rightarrow 1-0.4200 \\ &\Rightarrow {0 . 5 8 0 0} \end{aligned}$

The probability the elevator is overloaded is 0.5800

No, there is a good chance that 8 randomly selected people will exceed the elevator capacity.