Example 2 referred to an elevator with a maximum capacity of 4000 lb. When rating elevators, it

step 1 In this problem, we're told that an elevator is overloaded if it exceeds 5 ,000 pounds for its 2

Example 2 referred to an elevator with a maximum capacity of...

Example 2 referred to an elevator with a maximum capacity of $4000 \mathrm{lb}$. When rating elevators, it is common to use a $25 \%$ safety factor, so the elevator should actually be able to carry a load that is $25 \%$ greater than the stated limit. The maximum capacity of $4000 \mathrm{lb}$ becomes $5000 \mathrm{lb}$ after it is increased by $25 \%$, so 27 adult male passengers can have a mean weight of up to $185 \mathrm{lb}$. 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 $185 \mathrm{lb}$. (As in Example 2, assume that weights of males are normally distributed with a mean of $189 \mathrm{lb}$ and a standard deviation of 39 lb.) Does this elevator appear to be safe?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, often used as a model for natural phenomena. In this context, it is assumed that individual weights follow a normal distribution, allowing for the use of standard probability methods.

Sampling Distribution and Central Limit Theorem

When independent samples are drawn from a population, the distribution of the sample means tends to be approximately normal regardless of the original distribution if the sample size is sufficiently large. This is known as the Central Limit Theorem, and it is key in analyzing the mean values of groups, as in the scenario with multiple passengers.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is computed as the population standard deviation divided by the square root of the sample size. This concept is critical for quantifying the variability or uncertainty associated with the mean of a sample.

Z-Score and Probability Calculation

The z-score is used to translate a raw score into the number of standard deviations it is away from the mean. Using the z-score, one can calculate the probability that the sample mean exceeds a certain threshold by referring to the standard normal distribution, which is central to determining risks or chances in statistical assessments.

Safety Factor in Engineering

A safety factor is a multiplier applied to design limits to ensure that structures or systems can handle loads or stresses that exceed those expected under normal operation. This concept is crucial in engineering as it provides a margin of safety to account for uncertainties and variability in real-world applications.

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