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Sal ine recorts Ol nonieowners &nC the mean loss from fire in a year is H S500 per Ise and that the standard deviation of the loss is $10,000. (The distribution...

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Sal ine recorts Ol nonieowners &nC the mean loss from fire in a year is H S500 per Ise and that the standard deviation of the loss is $10,000. (The distribution of losses is extremely nt-skewed: most people have SO loss; but a [ew have losses The company plans to sell fire insurance for 0O plus enough to cover its costs and profit. Explain clearly why it would be unwise tO sell onlys policies. Then explain why selling many thousands of Ich policies is a safe business_ sells the policies for

Sal ine recorts Ol nonieowners &nC the mean loss from fire in a year is H S500 per Ise and that the standard deviation of the loss is $10,000. (The distribution of losses is extremely nt-skewed: most people have SO loss; but a [ew have losses The company plans to sell fire insurance for 0O plus enough to cover its costs and profit. Explain clearly why it would be unwise tO sell onlys policies. Then explain why selling many thousands of Ich policies is a safe business_ sells the policies for S600. )) Suppose the company 50,000 policies, what is the f the company sells probability that the average loss in a year pproximate Rrbe greater than S600? In 2005, the Federal 39 Weights of airline passengerd: updated its passenger Aviation Administration (FAA) of 190 pounds in the weight standards to an average 'ineludes clothing and '"(195 in the winter) This summer however; did not specily carry-on baggage: The FAA reasonable standard deviation is standard deviation_ Normally distributed , especially 35 pounds: Weights are nol men and women, but population includes both when the commuter plane carries they are not very non-Normal. probability thal, What is the approximate 25 passengers_ weight of the passengers exceeds in the summer; the total apply the central limit theorem; 5200 pounds? (Hint: To problem in terms of the mean weight) restate the aemia and physical 40 Iron depletion without studies have shown a link performance: Several



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The Ethan Allen tour boat capsized and sank in Lake George, New York, and 20 of the 47 passengers drowned. Based on a 1960 assumption of a mean weight of 140 lb for passengers, the boat was rated to carry 50 passengers. After the boat sank, New York State changed the assumed mean weight from 140 ib to 174 lb.
a. Given that the boat was rated for 50 passengers with an assumed mean of 140 tb, the boat had a passenger load limit of 7000 lb. Assume that the boat is loaded with 50 male passengers. and 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). Find the probability that the boat is overloaded because the 50 male passengers have a mean weight greater than 140 lb. b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2436 lb. If 14 passengers are all males, find the probability that the boat is overloaded because their mean weight is greater than 174 lb (so that their total weight is greater than the maximum capacity of 2436 lb). Do the new ratings appear to be safe when the boat is loaded with 14 male passengers?

Okay, we are going to answer question number 25 in your textbook water taxi safety. When a taxi, When a water taxi sank in the Baltimore harbor, an investigation revealed that safe passenger load for the water taxi was £3500. It was also noted that the mean weight of a passenger was assumed to be 100 and £40 assume a worst case scenario in which all the passengers are adult men. This could easily occur in a city that hosts conventions in which people the same gender often travel in groups, Assume the weights and men are normally distributed with a mean of £182.. a standard deviation of £40.8. So we have a mean of 182.9 And the standard deviation of 40.8. So you can see there's a lot of variability in this data set. I'm going to plot the distribution part A says, If one man is randomly selected, find the, find the probability that he weighs less than £174. Uh This is a new value suggested by the National Transportation and Safety Board. The probability that he weighs less than £174. I'm looking for the value to the left of 1 74 it's It's 41.3 per 7%. So a pretty small probability um that 174 might need to be re evaluated by the safety board. Part B says with a low limit of £3,500. How many male passengers are allowed. If we assume in me in a week of 140. So all we need to do is take 3500 And divide it by 140. And we could see that 25 passengers would be allowed on the boat. Uh Part C says with a lower limit of £3500. How many male passengers are allowed if we use a new mean of 182.9. So 3500 Divided by 1 82.9. Yeah. Is 19.1361. So approximately 19.1361 passengers would be allowed on the boat. Party says, why is it necessary to periodically review and revise the number of passengers that are allowed to board? Uh Well you can see that, you know, the old standard was 100 and £40. Unfortunately, people's weights have tend to be increasing over time. Uh So they assumed 100 and £40 per person would be good enough and they saw that that was too much. So they should re evaluate it since the um weight of people is constantly changing and they might want to re evaluate this periodically

The F A A. Gave information to the airlines at the average weight of about the average weight of passengers and in the summer, and what they said is that the average weight of passengers in the summer have a mean weight of 190 but they didn't give any cedar deviation. But a standard deviation that would be reasonable to consider is £35. And I know that my distribution is not normal because it contains both male and female weights. But it's not too far away from normal. So I'm doing my population. It's gonna be average passenger weight in the summer, and we're giving that The mean is 100 £90. And then we estimated that a reasonable standard deviation was £35. So this distance right here is £35. So part a. Ask us why you can't calculate the probability of one selected passenger that weighs more than £200. And the reason why that we can't calculate the probability for one randomly selected passenger is because our population data is it normal and you can't use it to calculate probability. If it's not normal, then part B says find the probability of the total weight of passengers on a full flight is greater than or equal to £6000. Okay, so they give the information about a commuter flight where 30 passengers. So this is a computer commuter plane. There's 30 passengers, and I know that the mean of this is going to be the mean of the population. So that's gonna be 100 and 90 and I need to find the standard deviation. So if there's 30 passengers, I know that in is equal to 30. So when I go to find my standard deviation, it's the standard deviation of the population, which in this case is 35 over the square root of the sample size, and in this case it's 30 so that's gonna equal to 6.39 So my standard deviation is about 6.39 Wants to intervene ation distance to the one standard deviation mark is approximately £6.39. Now. If I took take a look at the question. My question says it's the total weight of passengers on the full flight, and I know that there are 30 passengers on the flight in the hole. It needs to be greater than £6000. So if I take £6000 and divided by the 30 passengers, I'm really looking for an average weight of the passengers to be 202 £100. So, in reality, I'm looking for the probability that marine dimly selected passenger is going to be greater than or equal to £200. So what I need to do IHS, there's 1 90 once injured. Aviation is around 91 96. So 200 it's gonna be passed one standard deviation. So right here is where I'm estimating 200. Then I need to get what the standard deviation is for 200. So I'm gonna use mine Z score. So if I used my Z score, I'm gonna have the number of looking for just 200 minus the mean, which is 1 90 divided by the standard deviation, which is 6.39 and I get a Z score of 1.56 You know, I'm looking for the probability than it's greater than 100. So greater than 100 is going greater than 200 is going to be this probability. But when I look up the Z score of 1.56 it's actually going to give me a probability to the left. So to get the probability that I'm looking for harmony, have to look for the probability four Z is greater than or equal to 1.56 You remember the area under the total curve ISS one. So you're going to have to look for one minus the probability that Z is going to be less than or equal to 1.56 So that's gonna be one minus. I'm gonna look up 1.56 in the table and I get 0.94062 and that's going to equal zero point, he wrote five 938 And that's the probability that one person, the sample it's greater than are equal to £200 for the In reality, I know that there is basically a 6% chance that the total weight of the past teachers on a full flight is greater than people to £6000

So if the reality is true that the main weight of man Is 188 .6 lbs. With a standard deviation of £38.9.. And in part a they asked, well what's the likelihood that I randomly chosen male will weigh less than 100 and £74. And let's convert that to A Z. We have 174 -188.6, Divided by the 38.9 and left parentheses. E. 1 74 minus 1 88.6. Close my parenthesis E and then 38.9. And when I type that in I find that that gives me a Z value of negative .375 32 etcetera. And if we were looking this up in a table we would probably go to negative 0.38 and look that value up. I'm going to use software, my normal CDF button so I'm going to have a negative Say 1000 be my low limit. And then I'm going to for my upper use this value. And so it hits second in that button And I'll leave the mean and standard deviation and 01, respectively. So when I do that, I get that probability is .35 3 7. So about 35 chance of randomly choosing a male and be weighing less than £174.. Now, part B says if we assume that the mean was what missiles for males? If we assume that the main weight of a person on that, including males and females, and if the boat has a limit of £3500,, how many people we say we can get on there? And so if we divide, it looks like we can get 25 people on the boat. If their averages 140. On the other hand, if it were have your people or all males with this mean, and we divide, look how it is significantly less People. It comes out to 18.55, which I would say, let's round down and so we can only have 18 people Because if we round up, then you're gonna have to wait more than 3500 with that average. So there's a significant difference. So we definitely need to look at and review this. We need to review regularly to look at what the mean. Wait. Also, they may want to do some type of a look at gender and see which gender comes or just find some weights that are average and look at males and females about how many people get on the on the boat and so on. To make a judgment as to what that means would end up being.

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


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