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Serum Cholesterol As reported by the U.S. National Center for Health Statistics, the mean serum high density lipoprotein (HDL) cholesterol of males 20 to 29 years old is $\mu=47 .$ Dr. Paul Oswiecmiski wants to estimate the mean serum HDL cholesterol of his $20-$ to 29 -year-old male patients. He randomly selects 15 of his 20 - to 29 -yearold patients and obtains the data shown. Assume that $\sigma=12.5$ (TABLE CAN'T COPY)(a) Use the data to compute a point estimate for the population mean serum HDL cholesterol in Dr. Oswiecmiski's patients.(b) An analysis of the data indicate that the sample data come from a population that is approximately normalIy distributed with no outliers. Construct a $95 \%$ confidence interval for the mean serum HDL cholesterol for all Dr. Oswiecmiski's $20-$ to 29 -year-old male patients. Interpret this interval.(c) Do Dr. Oswiecmiski's patients appear to have a serum HDL different from that of the general population.(d) If Dr. Oswiecmiski wanted a more precise confidence interval, what would you recommend?

Intro Stats / AP Statistics

Chapter 9

Estimating the Value of a Parameter Using Confidence Intervals

Section 1

The Logic in Constructing Confidence Intervals about a Population Mean where the Population Standard Deviation Is Known

Confidence Intervals

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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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