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11. Serum cholesterol of 16 to 19 year old females approximately normally distributed with mean 171 and standard deviation 40 a. Determine the proportion of 16-19-y...

Question

11. Serum cholesterol of 16 to 19 year old females approximately normally distributed with mean 171 and standard deviation 40 a. Determine the proportion of 16-19-year-old females having serum cholesterol level between 150 and 200.b. Suppose a 16-19 year old female is randomly selected. Find the probability her serum cholesterol is less than 140.Determine the serum cholesterol that divides the bottom 10% from the top 90% of all serum cholesterol levels of 16-19 year old females:National Center f

11. Serum cholesterol of 16 to 19 year old females approximately normally distributed with mean 171 and standard deviation 40 a. Determine the proportion of 16-19-year-old females having serum cholesterol level between 150 and 200. b. Suppose a 16-19 year old female is randomly selected. Find the probability her serum cholesterol is less than 140. Determine the serum cholesterol that divides the bottom 10% from the top 90% of all serum cholesterol levels of 16-19 year old females: National Center for Health Statistics, 25th percentile of serum d_ According to 16-19 year old females is 145. How close is the 25th percentile cholesterol for of the normal curve to the above reported value?



Answers

Serum Cholesterol As reported by the U.S. National Center for Health Statistics, the mean serum high-densitylipoprotein (HDL) cholesterol of females 20 to 29 years old is $\mu=53 .$ If serum HDL cholesterol is normally distributed with $\sigma=13.4,$ answer the following questions: (a) What is the probability that a randomly selected female 20 to 29 years old has a serum cholesterol above $60 ?$ (b) What is the probability that a random sample of 15 female 20 - to 29 -year-olds has a mean serum cholesterol above $60 ?$ (c) What is the probability that a random sample of 20 female 20 - to 29 -year-olds has a mean serum cholesterol above $60 ?$ (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) What might you conclude if a random sample of 20 female 20 - to 29 -year-olds has a mean serum cholesterol above $60 ?$

All right, so we got an average of 206 and a standard deviation of 44.7, we're looking at all values between 152 50 so that's where our X. Is gonna go. So visibly we substitute these right here And we get negative 1.25. So 1.25. Standard deviations below the mean and just below one standard deviation above the mean. For 2 50 their corresponding percentages, as found in the back of the P Table two, it's not p values yet Um is 73.09% difference. Now we're looking for all those that are below 220 and we find that that is a Z score of almost a third. And when we look it up in table two it has a percentage of 0.6217. And now for the next part we're looking at uh oh the quantum is my favorite, so let's give me a 25% 50% and 75% respectively. Those are going to be negative 0.670 and 67 or 0.67 and X is just going to be given by the average plus the Z score times the standard deviation. So uh x at 0.25 Is going to be equal to the average 206 Plus negative 0.67 Times 44.7. And we get 176.051. And then the x of 0.5 or 50 Is going to be 206 because this thing is zero at 50 And that X is 0.75, We get 235.9 49 Yeah, So 25% of all observations are less than this. Number. Half of all the observations are less than 206 and half of the observations are three quarters of the observations are less than 235 point number 49 Mhm. And then finally we're looking at 40% of the table Or 44% of the graph which we can look up in the table. So he got her Z curve. Actually that doesn't look like a Z curve. Uh huh. Mhm. We're simply looking at the for fissile. So that's gonna be uh 50%. There's a 40% we're looking at this part right here And this has a corresponding z value. The value of 0.4 Is equal to negative 0.25. And then as before we can find its X. Value X value turns out to be 194.8- five. That's 40% of all observations are gonna be less than this number

Problem Number 35. Uh, you know, from the given that this confusion off em Ah, it's normal distribution with mean mu novem when equal toe won it eight and standard deviation. Okay, I am equal for keeper distribution. Off B is a normal distribution with mean be equal to 1 70 and stronger. And the Asian, um, equal to 30. This is from a given also ah, to calculate and the mean off a minus B is equal to mean and and minus me in off I'll be students sequence one 88 minus 1 72 is equal. Standard region off a minus B is equal to a square root off and the valiant and M minus close. And the radiance Well, you we know from the given that the central region 8 30 and 41. So the values is a squared with us a central division. So it's equal to the square walls. Ah, 41. Ah, it's weird. At plus 30 b s were was The final answer is 50 point point 0.0.8. This is a some abbreviation and the mean off a minus minus being Ah, this is ah so m minus B is a normal distribution with mean equal 18 and the Sunday the vision s standard deviation equal 50. And this is the answer for 35 35 8 certified B. We need to calculate the probability we need to calculate the probability that m minus B. It's more than other than you in order to use the value or the centralized very or to get the area under the curve, we have to calculate as a saint value which is equal on which is equal value the value When is the mean over the sunder the visa? We know that the very is zero. So is this zero You you know that the music will 18 and the sons of the religion is equal 50.8 with big question will be as u minus 18 over the center of the unity 15 points 0.8. So the final answer for that will be need to point 35. So why you think the tables with that equally limited open 35 we can read the final answer. That's the probability that a minus peace more than theory is equal to a one point three six three to the and

We're talking about the normal curve, and right now we want to talk about the relationship between any given normal curve and the standard normal curve, which is a super, super important distribution for all of statistics. So in this problem we are given a population, and that population is women over 20 years old, and we're told that the cholesterol in that population, uh, is a random variable that has distributed normally. So instead of cholesterol in women over 20 I'm going to just write X. That's going to be a random variable, and I'm going to write that is distributed normally by doing that little squiggle in the letter n. And then I'm going to write our parameters of our normal distribution. The question tells us that it has a mean of 206. I mean, is equal to 206 and our standard deviation is equal to 44.7. So now we have our normal distribution Now, this is not our standard normal distribution. This is just a normal distribution because it's mean is not one and its standard e. It means not zero, and the standard deviation is not one So the first part of this question wants us to draw our distribution. And we can do that pretty easily. I'll draw my X X is down here. This isn't going to be perfect. Um, but we do know a couple of things about our, uh what the normal distribution looks like. It's going to be centered over our means. Alright. Two or six right here. And as we go further from the mean, it's gonna go closer and closer to the X axis, so all sort of mark where our standard deviations are going to go. So we'll say this first tick mark is one standard deviation away from the mean, which is going to be about 250. The second will be two standard deviations from the main between me, about 294. Of course, I'm truncating This isn't going to be exact. And then I think 338 is three standard deviations away. I could be doing my bathroom, though. Now, in the other direction, we're going to have 1 62 and then, uh, 1 60 to minus 44 is 1 20 to minus four, which is one 18 and then one more standard deviation to the left will have, Um, let's say 1 14 minus 40 is think 1 74 again. I could be doing my math wrong is always in my head. And so we know a good rule of thumb is that once you're three standard deviations away from the main, um, the normal, the normal curve is basically at the X axis. It's not exactly, but it's close, Um, so we'll just sort of draw it like that. And that's what our curve is going to look like our normal curve. This is the distribution of X, our cholesterol levels in women over 20. Now this curve is because we had to do a lot of sort of estimation guessing about where this curve is going to land, what it's going to look like. Um, so it's not super easy to work with. There's an easier curve to work with, and that's the standard normal curve. Now the standard normal curve. That's a distribution, a normal distribution with a mean, equal to zero and a standard deviation equal to one. And you can see with numbers like that that distribution is going to be a lot easier to work with, especially when we need to do calculations of area and things like that. So how can we translate our normal curve to this standard normal curve? Well, we have a formula for that. And if we want X to transform into Z Z is the random variable that we use for our standard normal curves. Z comes from the standard normal curve. All we need to do is subtract the mean from X and then divide it by the standard deviation. So in this instance, we're going to have X minus 206 all over the standard deviation of 44.7 and that is our ZR standard normal, random variable. Now, if we want to draw Z now, it's gonna be a lot easier. So part CSS to sketch a drawing of Z. And we know exactly what this is going to look like. We have zero here in the middle and I'll do one, 23 and on the other side, negative one negative two and negative three. And now we will sort of just sketch what this is probably going to look like. It's going to be something like this again. Not perfect, but with the standard normal curve, we get a better idea. We know more about what this is going to be. So now we get down to the meat of this question and we're and we're asked if we know the percentage of all outcomes between with cholesterol levels between 1 52 50. So if we know the percentage all outcomes between 1 50 2 50 what if that's what we're trying to find out? Excuse me, we don't know that, but we want to find out what the percentage is. Where, on the standard normal curve are we going to have to look? So what do we know about this phrase? Percentage of all outcomes within a range. We know that is the same thing as saying the area under the density curve of our variable. So in this case, it's X under density curve of X between the same limits of 1 52 to 50. Now, hold on. That's not going to be very easy to find. We have a density curve of X. We know it's mean and standard deviation, but unless we have a calculator or some computer program handy. It's not going to be very easy to find there's one curve that we have a table of values because the standard curve and that's the standard normal curve. So if we were, if we're able to translate this area into an area relative to the standard normal curve, then will be much more easily able to figure out, um, what this percentage of all outcomes is well to do that we just need to translate these values for X into values for Z. So if we are rewrite, this is going to be equal to the area under the standard normal curve and when we have X is equal to 1 50. C x equals 1 50 z is going to equal 1 50 minus the mean, which is 206 divided by the standard deviation, which is 44.7. And we'll find that that is a value of negative 1.25 to 8. And when we have X is equal to 50 Z is going to be 2 50 minus 40 minus two of 6/44 20.7, and so we'll have a value of 0.9843 And so these are the limits that we're looking at for the percentage of all outcomes when we are finding the area of the standard normal curve. And why is this helpful? It doesn't look like this is going to be helpful, because now we're dealing with a bunch of messy decimals instead of some round numbers. Well, now this is the standard normal curve. We have a table of values. It's right in the back of your textbook. It's very easy to find that we can look for the area to the left of this when in the standard normal curve, we can subtract that we can subtract from that area to the left of this, and we'll find the area in between these two, which is going to be the same as the percentage of all outcomes from 1 52 to 50. Now the final question is very, very similar. Asks just a different form of this question. If we want to find the percentage of all outcomes to the left of 220 where do we need to look on the standard normal curve? Well, it's going to be the same exact thing. We need to translate this standard normal or this X into a Z and by doing and to do that we do Z is going to equal 220 minus Sarmiento Oh six and divided by the standard deviation 44.7. So we'll get 0.3 1 32. And since we're looking at all the values for cholesterol that are less than 220 we want to look to the right or takes you not to the right to the left. We want to look to the left of 0.3132 So that's what our final answer is going to be if we have our standard normal curve here and here, of course, is zero. We're going to look a little bit to the right of zero to find 0.3132 and then our percentage of all total outcomes with cholesterol less than 220 is going to be the same as the area here to the left of 0.3132 and these are your final answers

The following is a solution video. It's number 29 and this looks at dr I'm gonna butcher this name. But the doctor um Asswipe Muskie's 20-29 year old patients, they're mean serum HDL cholesterol level And were given a data set. Now it's hypothesized that the mean is 47 that from previous studies the population standard deviation is 12.5. So we're assuming actually in fact it even says we're assuming normality here so we can use the z interval because we know the population standard deviation. We actually know what sigma is. Whenever you know what sigma is. You can use the Z interval. So the first part is to find the point estimate. Now I use technology to do that but you can use the formula if you so wish. But if you have a T. I, 84 83 you go to stat and then edit and you can see here the data values here. So I don't know if you It's all those but there's there should be 15 data values I think they were 15. Which still check real quick 15 data values. And if you go to stat again and then air over the couch and then you click on one of our stats. Now I put my list in L. One so I'm gonna keep that as L. One And then you press calculate and that X. Bar that's your point estimate for the main. So 48.33 three repeating I guess 48 3rd. So 48 we'll just say 48.3 is the point estimate. Okay so now we gotta find the 95% confidence intervals. So again I'm going to turn to the calculator because it's a lot quicker this way. But you can certainly use the formula and we go to tests and it's the seventh option here. Like I said we can use the Z interval whenever we know what sigma is. Um And we are working in data here so we're going to keep the data up here highlighted for input. So it's not summary stats, it's actual data set. The sigma, the standard deviation was 12.5. The list was L one at least. Mine was I put it in a one frequency should be one and then the sea level, the confidence level remember was 10.95 or 95%. And then we calculate in this top band here that's our confidence and also 42.008 All the way up to 54.659. Let's go and write that down and then we'll interpret it so 42 .008 two. points 659 Okay? And then as for the interpretation will say we can be 95% confident that the mean serum H. D. L. Cholesterol for all Dr I'm not going to try to press this. Yeah I can't even feel it right. I don't know who it is but someone smart on me. So dr owes 2022 29 year old patients is between 42.008. And you can round differently if you want. I'm not gonna go in and around And 54.659. I'm not 100% sure on the units here. I'll just put HDL. Okay and then the part c um It says is that different from the hypothesized value of 47? And it's not because, so no because 47 lies within that range. In fact it's almost exactly right. So it's it's pretty darn close. I mean it's right in the middle of that confidence interval, but as long as it's contained in that confidence interval, we can say that, you know, there is no difference from our confidence interval from the previous study, so No, Because new equals 47 is contained. And the interval. Okay. And then part D. A says, what would you do to be more precise? The best thing, there are a few things you can do at least to make it narrower, the confidence that were narrow. So you can kind of like Shorten that up. So maybe instead of between 42 and 54, if you want to get it to maybe like, you know, 44 and 52 or something if you wanted to have a narrower interval. Um the best thing to do is just increase your sample size to be more precise. Ah I would say this is kind of a professional but I would say I would increase the sample size. The n. That's probably the best thing you can do if possible. Now, if it's not possible. The other thing you can do um If you wanted to narrow that confidence interval down, you could say or decrease confidence level. Now this is where it gets a little bit hazy because you know, what do we mean by more precise? Do we need mean like we want to be as exact as possible? Because if that's the case then we would want to increase the confidence interval confidence level, but that's going to make it wider. So I don't know if that's, you know, 100% precise. If you wanted to make it a narrower interval, which I think that's what they mean by being more precise, then you would need to decrease that confidence that. But to make it, you know the best answer here really, you need to increase your sample size. If you want to be more precise, that's the best thing to do.


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