Elevator Safety Example 2 referred to an elevator with a...

Elevator Safety Example 2 referred to an elevator with a maximum capacity of 4000 ib. 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 ib becomes 5000 ib after it is increased by $25 \%,$ so 27 adult male passengers can have a mean weight of up to 185 ib. 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 ib. (As in Example $2,$ assume that weights of males are normally distributed with a mean of 189 ib and a standard deviation of 39 Ib.) Does this elevator appear to be safe?

Key Concepts

Sampling Distribution of the Mean

When a sample is drawn from a population with a known distribution, the distribution of the sample mean also follows a normal distribution, provided the sample size is sufficiently large, even if the underlying distribution is not normal. The standard deviation of the sample mean is the population standard deviation divided by the square root of the sample size, which helps in assessing the variability of the mean across different samples.

Probability of Overload

The probability of overload is an assessment of the likelihood that the actual load or measurement exceeds a defined threshold, taking into account the inherent variability in the measurements. This concept is critical in safety analysis where, by using the normal distribution and sample statistics, one can estimate the risk that a system—like an elevator—experiences loads beyond its safe designed capacity.

Safety Factor

A safety factor is a design principle that includes a margin of error by increasing a system’s nominal capacity to reliably handle unexpected overloads or uncertainties. In engineering, adding a safety factor means that the designed load capacity is higher than the predicted maximum load, ensuring additional security against potential failures.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is commonly used in statistics to represent real-valued random variables whose distributions are not known, particularly when dealing with natural phenomena such as human body weights.

Z-score and Standardization

A Z-score is a statistical measurement that describes a value’s relation to the mean of a group of values, measured in terms of standard deviations from the mean. Standardization, which involves converting individual sample observations into a Z-score, enables the use of the standard normal distribution to calculate probabilities and assess the likelihood of a particular outcome.

Transcript

00:01 This problem is all about elevator safety.

00:05 And let's say we have the elevator.

00:08 And the elevator has a maximum capacity rating of 4 ,000 pounds.

00:16 And we always like to build in a safety net.

00:19 So even though it says 4 ,000 pounds, it's pretty common to use a 25 % safety factor.

00:26 So if i calculate 25 % of 4 ,000, i'd get 1 ,000.

00:31 So if i add on that thousand, the 4 ,000 rating, even though it says 4 ,000, can technically hold 5 ,000 pounds without having issue.

00:43 So now let's say we are trying to put 25 adult male passengers into that elevator.

00:49 So if i take 27 passengers and i divide it into the 5 ,000, i'm going to get an average male of about 185.

01:02 So anything over 185 pounds would be classified if all the people were over that average weight, then my elevator would be overloaded.

01:16 So the question that is being asked here is to find the probability that it's overloaded because they have a mean weight greater than 185.

01:29 So putting that into symbol notation, we're looking for the probability that the mean weight is over the 185 pound mark.

01:40 In doing so, there's a couple assumptions that have to be made, and they're provided to us.

01:47 We are assuming that the weight of males are normally distributed.

02:18 We are also given some information that the average male is 189 pounds, with a standard deviation of 39 pounds.

02:41 So using this information, we want to determine the probability that the average is greater than 185.

02:49 In order to tackle this problem, we are going to draw our bell -shaped curve, and we're going to have to talk about sample means.

03:01 So our sample in this case is we're putting 25 people on the yellow, or sorry, 27 people on the elevator.

03:11 So our sample size is 27.

03:15 And we're going to need to discuss the average of the sample means and the standard deviation of the sample means.

03:24 And the average of the sample means and the standard deviation of sample means can be found utilizing your central limit theorem.

03:32 And the central limit theorem says the average of the sample means will equal the average of the population which in this case was 189.

03:41 We got it from here...

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