Ergonomics. Exercises 9–16 involve applications to ergonomics,...

Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.

d. Is the new capacity of 20 passengers safe?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A small commuter airplane has a weight limit of 2400 pounds for passengers. If the average weight of adult passengers is 165 pounds, with a standard deviation of 30 pounds, is it safe to allow 12 adults on the plane? Awesome. So it appears for this particular problem we're asked to determine whether or not it is considered safe to allow 12 adult passengers on this specific small commuter airplane. So yes, no. Etc. So with that in mind, now that we know that we're trying to determine whether or not it's safe to allow 12 adults to ride on the small commuter airplane, let's quickly read off our multiple choice answers to see what our final answer might be. So A is yes, it is safe, B is no, it is not safe. C is safe only for short flights, and D is cannot be determined. Awesome. So with that in mind, with that said, our first step in order to solve this particular problem is we need to identify our given values, so we need to write down all of our known variables. So we're told that the weight limit for the passengers aboard this plane is 2400 pounds. So which pounds is is denoted as LBS. Fantastic, or LB for short, so for pounds. So, once again, our weight limit is 2400 pounds, and the average weight of an adult passenger is 165 pounds. Fantastic. That's our average weight limit, or we have our average weight limit and the average weight of an adult passenger, and we're also told the So once again, to be more specific, since we know that the average weight of an adult passenger is 165 pounds, this is equal to our mean, which is denoted as me. So that's our population mean value, and then our standard deviation, which we'll call sigma, and sigma, once again, our standard deviation of the weight of an adult passenger, as we're told by the prom itself, is 30 pounds. Fantastic. And we're also told that the number of adult passengers, which we'll call this lowercase n, so the number of adult passengers is equal to 12. So now, our next step is we need to calculate the sample mean weight limit for each adult. So, as we should recall a note, this will be the total weight limit divided by the number of adults that are allowed on the plane. So we'll call this equation X bar, and X bar. is equal to 2400 pounds divided by 12, which is going to be equal to, when we plug into our calculator, 200 pounds. So now, our next step is we need to calculate the standard error of the mean, which we'll call this value sigma subscript X bar. And as we should recall, this is calculated by dividing the standard deviation by the square root of the sample size, so that will mean that Sigma subscript X bar is equal to 30 pounds, and then we need to take 30 pounds, and when you divide it by the square root of 12, so when we plug that into our calculator, we will get 5 multiplied by the square root of 3 pounds. So now, our next step is we need to calculate the Z score, and as we should recall, this tells us how many standard errors the sample mean is from the population mean. And since we are comparing the average weight of 12 adults, which is 200 pounds, to the average weight of an adult passenger, which is 165 pounds, we can therefore calculate, as we should recall and note, the Z score by recalling and using the following formula for the Z score, which the the Z score is equal to X bar. Minus mu, all divided by sigma subscript X bar. Fantastic. So when we plug in our known variables, we will find that this is equal to 200 minus 165 divided by 5 multiplied by the square root of 3. So 5 multiplied by the square root of 3. So when we plug that into a calculator and we round to two decimal places, we will have and we'll find that we'll get an answer of 4.04, fantastic. So now, our next step is we need to use the Z score to help us determine the probability that the total weight of 12 randomly selected adult passengers will exceed the 2400 pound weight limit. So we need to take P of or capital P denotes the probability. So P of capital Z, so Z is greater than 4.04. Awesome. So P of Z is greater than 4.04, which is gonna be equal to 1 minus P of Z is less than 4.04. Fantastic. So we now need to look up the Z score of 4.04 in a standard normal distribution table, which you should have a standard normal distribution table in your textbook. So when we use. The standard normal distribution table from your textbook, you will find that P of Z is greater than 4.04 will be equal to 0.9999, so 49. So therefore, as a result, the probability that the total weight of 12 randomly selected adult passengers exceeding. This 2400 pound weight limit will be P of Z is greater than 4.04 is equal to 1 minus 0.9999, which is gonna be equal to 0.0001. So this will mean that there is a 0.0001 chance that the total weight will exceed 2400 pounds, and since this probability is extremely small, As a result, because it's extremely small, it is very safe, or it's considered very safe to allow 12 adults on board this small commuter plane, based on the average weight and the standard deviation that is provided to us by the prom itself. So that will mean when we look at our multiple choice answers, the correct answer has to be multiple choice answer, letter A, which is yes, it is safe. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Normal Distribution

Mean and Standard Deviation

Load Capacity and Safety Assessment

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