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Problem 4A rare disease affects one in every one million people on average in the US. A national public health organization investigates large sample of people in t...

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Problem 4A rare disease affects one in every one million people on average in the US. A national public health organization investigates large sample of people in the US. Assume all people inves- tigated are independent of each other in terms of this disease_ In each of the questions below, select THE SINGLE correct statement.questionsand (b), please pick the most appropriate of the multiple-choice answers.Assume the people are investigated one after another: Let X be the number of people invest

Problem 4 A rare disease affects one in every one million people on average in the US. A national public health organization investigates large sample of people in the US. Assume all people inves- tigated are independent of each other in terms of this disease_ In each of the questions below, select THE SINGLE correct statement. questions and (b), please pick the most appropriate of the multiple-choice answers. Assume the people are investigated one after another: Let X be the number of people investigated until the organization finds the first person with the disease. The average and standard deviation of X are about With Xas in (a), the chance that the organization has to investigate 2000000 people before finding their first person with the disease is In a group of 10 people chosen at random by the organization, the probability that at least three people have the disease is equal (d) In & group of 200,000 people chosen at random by the organization, the number of peo ple in the group with the disease is best modeled as With the situation as in (d), the number of people in the group with the disease has an expectation and standard deviation respectively equal to



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Suppose that 20$\%$ of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions.
(a) What is the probability that when the experiment terminates, four individuals have not had adverse reactions?
(b) What is the probability that the drug is administered to exactly five individuals?
(c) What is the probability that at most four individuals do not have an adverse reaction?
(d) How many individuals would you expect to not have an adverse reaction, and how many $\quad$ individuals would you expect to be given the drug?
(e) What is the probability that the number of individuals given the drug is within one standard deviation of what you expect?

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

It's a kind of an interesting question if we we would be assuming that the anxiety level if we are going from easy to difficult is equal to the anxiety level. If the questions were difficult to easy and alternately, we just want to know if there's a difference and easy to difficult and not equal to difficult to easy. And so we're going to assume that that difference is actually zero so that the, I'll just write E G minus D. E. That that difference is zero. And when I put this into my calculator, um we have the lists have different sample sizes. We have that first group of the E. T. D. I have, there are 25 numbers versus the D two E. There are 16 numbers and we could use the degrees of freedom of 15, but I'm going to use the degrees of freedom from the formula. And that degrees of freedom ends up being uh 30 almost 39. So 38 point well, I'm just going to call it 39 it's approximately 39 degrees of freedom. And that test statistic that we're going to get, we need to take the mean, which was 27.1152 minus the mean of the other group, which was 31.7 to 8125 And then divided by the square root of. And the first standard deviation all around that a bit is 6.857 one square, divided by the sample size, which was 25. And then the second standard deviation was 4.26 square divided by the sample size of 16. And when I got that test statistic, the test statistic came out to be, I'll just read it up here negative 2.6 566 And so it came out down here. And since we're doing a two tailed test, we also use this one. And so what's the likelihood of getting a test statistic in this distribution that is less than or equal to that negative 2.6566 That gives us this tale and then we want the other tales to double it. And that p value comes out to be a 0.114 So at a 5% significant level, this is smaller than 5%. So we would have sufficient evidence to reject now. Yeah. And say that the mean anxieties are different so that they're different. The means are different. However, at a 1% significance level we would fail to reject the novel. Mhm. And we'd have to say here that they're actually not. They appeared to be the same. So it does depend on our significance level. How did pick you want to be. But it is an interesting idea in all my years of teaching, I never thought about the arrangement of having them all go easy, too difficult or difficult to easy. So what interesting concept.

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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