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Numerade Educator

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Problem 13 Medium Difficulty

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.
a. Given that the water taxi that sank was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers?
b. If the water taxi is filled with 25 randomly selected men, what is the probability that their mean weight exceeds the value from part (a)?
c. After the water taxi sank, the weight assumptions were revised so that the new capacity became 20 passengers. If the water taxi is filled with 20 randomly selected men, what is the probability that their mean weight exceeds 175 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb?
d. Is the new capacity of 20 passengers safe?

Answer

a. 140 lb$\\$

b. 0.9999999998 (Table: 0.9999)$\\$

c. 0.9458 (Table: 0.9463)$\\$

d. The new capacity of 20 passengers does not appear to be safe

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Video Transcript

right. This problem is about water taxi safety and a scenario of an incident that happened in Baltimore's inner harbor. And there's Cem, given information that we have to start with before we can answer the four parts to this question. And the given information is that waits of men are normally distributed that the average weight of a man is £189 with a standard deviation of £39 and we were also told about the occupancy of the boat. The stated occupancy of the boat was 25 people, and the load limit was £3500. And keep in mind, those 25 people could have been men, women and Children. So when they do the stated occupancy, they have to talk in terms of both a weight limit as well as a person limit. So let's go to problem A and in problem A. It asks, given that the water taxi that sank was rated for a load limit of £3500. What is the maximum mean weight of passengers if the boat was filled with the 25 passengers? So if we take those 25 passengers and we divided into the 3500 possible Wait, We're going to end up with an average of 140 pounds again. Keep in mind, that's average. So that meant that some of the 25 people could have been weighing more than 100 £40 and some of them could have been weighing less than £140 in part B. The problem is asking you if the water taxi is filled with 25 randomly selected men, so we're going to focus on the heaviest class of people We've got the men, the women and the Children and usually the men. Average weight is higher than any women, average weight or child's average weight. So we're drawing a sample from our population and our sample size is going to be 25. And we're trying to determine what is the probability that they're mean. Wait exceeds, which means greater than the value from Part A, which we found to be £140. And in order to solve this problem, we are going to have to discuss the average of the sample means because again we are finding a sample of 25. We're talking about their mean wait and we need to discuss the standard deviation of those sample meets. And we're going to let the central limit theorem guide us in calculating these. And the central limit Theorem says that the average of the sample means will equal the average of the population, which in this case was £189. And the standard deviation of sample means will be equivalent to the standard deviation of the population divided by the square root of the sample size. So in this case, is going to be £39 divided by the square root of 25 and because they told us that it was normally distributed. We can start this process by drawing are bell shaped curve, and when we draw our bell shaped curve, we're going to put the average in the center, and we're trying to calculate the probability or the likelihood that the average is greater than 140. We will need a Z score formula to assist us in solving this problem, and since we're dealing with sample means the Z score formula that we're going to use is going to be X bar minus the average of the sample means it's that over a little bit all over. The standard deviation of the sample means. So we need to calculate R Z score for 140. So we're going to do Z equals 140 minus 189. And in place of the standard deviation of sample means we're going to use the expression 39 divided by the square root of 25. In doing so, you get a value of negative 6.28 So when we're solving this problem, if I put that negative 6.28 up here on my bell, when I discuss the chances of the average of thes 25 men's being greater than £140 it's no different than saying. What's the probability that the Z score is greater than negative 6.28? And because our standard normal table, which is found in Table 82 in the back of your book, always discusses theme area or the probabilities into the left tail of the bell, and as you see our picture is going or extending into the right tail. We're going to have to rewrite this probability as one minus the probability that Z is less than negative 6.28 So when we go to our standard normal table, um, the negative 6.28 is not found in the table. But you do have a statement in that table that says that a Z score that is less than or equal to ah, 3.5 or negative 3.5 is going to have a probability of 0.1 So when we subtract the one minus the 10.1 we're getting a probability of 0.9999 So let's recap Part B in part B. It's saying if the water taxi was filled with 25 randomly selected men, what is the probability that they're mean? Weight exceeds that £140 from part A, and that probability is going to be 0.9999 So let's move on to part C and in part c, you are asked. After the water taxi sank, the weight assumptions were revised. So the new capacity, instead of being 25 people, we now are only going to do 20 people. So now we're going to have a new sample size and this time our sample size is going to be 20. And if the water taxi was filled with 20 randomly selected men and again, we chose the men because the man's weight is usually heavier than the woman or the Children. If the water taxi is filled with those 25 randomly selected men, what is the probability that they're mean wait exceed, which is greater than £175? So again, we're going to need to calculate with this new sample, the average of the sample means and the standard deviation of the sample means. So again we can apply our central limit theorem. The average of the sample means is equal to the average of the population, which in this case was £189. As the average weight of men and the standard deviation based on the central limit theorem would be equivalent to standard deviation of the population divided by the square root of the sample size. And our standard deviation of the men's weight was 39 and our sample size here is going to be the square root of 20 again. We're going to draw our US bell shaped curve. We're going to put the average in the center, and we're trying to determine the probability that we are greater than 1 75. So again, the Z score we're going to use is going to be Z equals X bar, minus the average of the sample means divided by the standard deviation of sample means. And in this case, it's going to read to see equals 1 75 minus 1 89 divided by Here's that expression for the standard deviation of the sample means we're gonna have 39 over the square root of 20. And that's going to yield a Z score of negative 1.61 so we can put a negative 1.61 up here on our bell. And then when we're talking with our problem being, what's the probability that the average is greater than 1 75? It's no different than saying What is the probability that your Z score is greater than negative 1.61 and again, our our picture is extending into the right table tail. But the table in the back of your book discusses the areas or the probabilities as we extend into the left tail. So we're going to have to rewrite this as one minus the probability that Z is less than negative 1.61 And when you look that value up in the table, you're going to get one minus 10.537 for an overall probability of 0.9463 So recapping part C. After the water taxi accident, they downgraded the capacity number from 25 passengers on Lee, allowing 20 passengers. And when we put 20 men on that water taxi, the probability that their average weight exceeded £175 would be the 0.9463 And then finally, Part D. In this problem in part D, the question is saying, Is this new capacity of 20 passengers safe? Um, when you look at the new capacity, the new capacity of 20 passengers still does not appear safe because the probability of being over the load limit is still high. At a value of 9.9463