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A random sample of 50 female American health professionals yielded the following height data.$$\begin{array}{llllllllll}\hline 65 & 66 & 64 & 67 & 59 & 69 & 66 & 69 & 64 & 62 \\63 & 62 & 63 & 64 & 72 & 66 & 65 & 64 & 67 & 68 \\70 & 63 & 63 & 68 & 58 & 60 & 64 & 66 & 64 & 62 \\65 & 69 & 64 & 69 & 62 & 58 & 66 & 68 & 59 & 56 \\64 & 66 & 65 & 69 & 67 & 67 & 68 & 62 & 70 & 62 \\\hline\end{array}$$a. What population was sampled to obtain the height data listed above?b. Describe the sample data using the mean and standard deviation, plus any other numerical statistic(s) that help describe the sample.c. Construct a histogram and comment on the shape of the distribution. Construct any other graph(s) that help describe the sample.d. Using the statistics found in parts b and c, estimate the mean height of all American female health professionals using a single value. Using an interval.e. What quality of the interval estimate would improve the worth of the interval?

a. female health professionalsb. $\bar{x}=64.8, s=3.5$c. mounded about center, approximately symmetricald. approx. 65 inches; 62 to $69,$ the heights most frequent in this data.e. A narrower interval would be very desirable and/or larger sample size.

Intro Stats / AP Statistics

Chapter 8

Introduction to Statistical Inferences

Section 1

The Nature of Estimation

Hypothesis Testing with One Sample

Temira M.

July 8, 2022

(c) What can we conclude from the five number summary about the location of the 30 th percentile? The 55 th percentile?

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Okay, so in this question, we are given this data set, and I'm here on desk. Most, um, desk must Ah, come on. Their nice graphing calculator. So here I have rewritten all of the the data values they give us so that we can just reference this list X one. Um And so the first thing that asks for us to do is find the mean I'm so right down here. I just say mean of X one, and it gives me to gives it to me right here. 64.78 Um, and then the next thing is it once the standard deviation. So the command is S t d E B, and then you can just put your list and again, x one, It's our standard deviation is three point or six by eight. Um, now Ah, next thing it has us do is create this history, Graham. So I created this history, graham and preset kind of the settings here. But all you do is you type hissed a gram and then put an x one, and it'll create. And you might have to, um, go up into this top left corner and click on your settings menu to make sure you get all your data values. So I knew my lowest one is 56 on my highest one was 72. So I just did, like, 55 to 75 so that I could guarantee that I could see each of those values. Um, and then I also made sure that my count, it was just a little bit higher than, um, where it is right there. I'm in here. I have my Ben with that one. You could change your Ben Wittes. Um, especially if it looks very scattered. Your data looks very scattered or, um, looks like would be helpful to condense your data. However, weaken, weaken, see a pretty good trend here with with just the single Been with, um, an example of when you might want Teoh Change your Ben width is if you have, like, a bunch of values or sporadic values in here that are just, um, like, randomly a lot lower or something like that. But these these are pretty good values on so thin they ask us on the hissed a gram to describe the shape of the distribution. And so when I look at this distribution we have, um our peak is pretty well centered. It's not really far over to the right, not really far over to the left, Um, and it's pretty symmetric. It doesn't look like it drops off Aton on one side more than the other. Um, they're pretty close to even, um and so I would call us a symmetric distribution with one mound in the centre. Um, and so that's how we can describe it and then using using both the mean in the history Graham here to give an estimate. Um, using the history am it looks kind of like 64 but especially where it's a pretty symmetric pretty ah, and and that the pique is here in the middle, mounted in the center, we can usually use our mean, exactly, um, on skewed distributions. That's a little bit different, but where this is very symmetric and very mounted in the center, we can go ahead and use our mean, which also tells us it's a little bit above 64. So we might say, like, 65. Um, and that's not unrealistic. It's right here. It's right between these two large peaks. Um, so we might want to say, like, 65. And then if it then it asks us to describe it in a range. Um, and so one way to look at the range as well. Most of my values air right here in this big central grouping. So I might say the range from 62 to 70 just because almost everything's right there in that grouping. There's one out on outside side, and there's maybe five observations down below. Um, and so that's a pretty good way to do it. The history, Graham. Other people will do it with standard deviation. Um, in which case, you just add a really safe bet is like two standard deviations on either side of the mean, um, so my means here. And I might move, like, seven values this way and seven values that way for two standard deviations. That's a really safe standard deviation. Our estimate, um, one standard deviation is also okay. Um but for this particular one, where we see a lot of these could be an ally or something, I'd probably just use that 1st 1 62 to 70. Um, and and finally it asked us what what would we want to improve that. And really, these gaps are a big concern. We want to know. Are these really out layers? And so one of the best things that we could do to improve that estimate or make it more meaningful is if we had more data, um, to to make sure that we understood the whole story. Um, they also say we could use, like, a smaller gap. That's more meaningful because we're more precise. But then we we begin to have a little bit less uncertainty. But one thing that can always help is ISMM or a larger sample size or or a greater number of observations not always help us, um, secure. Ah, uh, our estimates.

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