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StudentQuizLectureTne 9970 confldence interval for tne dlfference mean pulse rates between students in class lecture and taking the quIz (class lectur pulse rates q...

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StudentQuizLectureTne 9970 confldence interval for tne dlfference mean pulse rates between students in class lecture and taking the quIz (class lectur pulse rates quiz pulse rates) (round t0 decima places}:TqueFalse The_probability that the true mean difference lies in9936 confidence interval you calculated is 0.99:We should analyse the data &s matched pairs procedure True or False:We mignt expect the mean pulse rates to increase under the stress of quiz Test whether the data provide suffici

Student Quiz Lecture Tne 9970 confldence interval for tne dlfference mean pulse rates between students in class lecture and taking the quIz (class lectur pulse rates quiz pulse rates) (round t0 decima places}: Tque False The_probability that the true mean difference lies in 9936 confidence interval you calculated is 0.99: We should analyse the data &s matched pairs procedure True or False: We mignt expect the mean pulse rates to increase under the stress of quiz Test whether the data provide sufficient evidence to support thls claim; State the value ofthe test statistic (round to decima places}: Whar thep-value of your test (round Io cecimal places)? Atthe 59 level of significance we reject True or False?



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A researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The data she collected are displayed in the box-plots and summarized below.
$$\begin{array}{ll}&{\text { Sex }} \\ & {\text { Male }} & {\text { Female }} \\ {\text { Count }} & {28} & {24} \\ {\text { Mean }} & {72.75} & {72.625} \\ {\text { Median }} &{73} & {73} \\ {\text { Median }} & {5.3725} & {7.69987} \\ {\text { Range }} & {20} & {29} \\ {IQR} & {9} & {12.5}\end{array}$$
a) What do the box plots suggest about differences between male and female pulse rates?
b) Is it appropriate to analyze these data using the methods of inference discussed in this chapter? Explain. create a 90$\%$ confidence interval for the difference in mean pulse rates.
d) Does the confidence interval confirm your answer to part a? Explain.

Yeah. Alright. In this question we are given the pulse rate of nine individuals in the class and we're told to consider them a population and part a asked us to find the population main. Mhm. So to find the population mean we're simply going to take all of the pulse rates And added up and divided by the total number, which is nine. Okay. And this gives us a mean pulse rate of 72.22 and it's in beats per minute. Then on part B they ask us to take three different um samples from the population and compute the sample means. So the first thing we're gonna need to do to do that is we're gonna need to number our population so that we can randomly choose three From the population. And what I did is I've used my t. I. 84 calculator um and told it to do random integer. No repeat. So I'm asking it to choose three numbers between one and nine without repeats. And here are my three samples. So my first one um was number nine. So I'm going to call this sample number one. So individual number nine had a pulse rate of 73. Number two had a pulse rate of 60 and number five had a pulse rate of 72. So I'm going to some those up and um divide Some that up and divide by three. And that's going to give me a sample pulse rate here of 68 .33 beats per minute. Then we're gonna do the same thing for my next sample. So for sample number two, I chose individual four individual seven. An individual number one same thing. We're going to add those up And divide by three. And that gives me 79 beats per minute. And then we'll go ahead and do our last sample. So sample three. I chose four and nine and two. Going to divide by three. Okay. And that gives us 71.3 beats per minute. And then it just asked us to think about um really we're just comparing the population. Um I mean, which was the 72.22 beats per minute. We want to compare that to the three different sample means we had and it just asked us to um to do that and to think about if we've overestimated or underestimated and this is part C. So overestimated was sample number two Because I had 79 which is higher than 72 and then underestimated. Um Well, both sample one and three underestimated. And then exact. None of my samples were exact. And again, you know, if you do this on your own, you're going to have three different samples. And so your answers may vary, but the sample means are somewhat close, but not exact.

All right. I have he 95% confidence interval from 70.8876 to 74.4970 What does this mean? This means that I am 95% sure where there is a 95% probability that the original mean off the population lies between these students lies between these two numbers. This is what this output means. Now, what is the margin of error? Margin of error is simply going to be the addition off these to develop by two. So it is 70 0.887604 plus 74.49701 One day we're but to end this actually turns out to be approximately 1.8. This is my marginal Federer. What is the sea question? The sea question says, if I want to calculate a 99% confidence interval what the margin of error be, it will be larger because now I want a fight arranged. So the margin of error will be larger. Sorry. These are my answers.

All right. So we are looking at pulse rates and we are going to be looking at this data and we're going to do some measures of the means of these pulses. So friday were to compute the population. Mean that all these pulses. So let's do a So the mean this the notation that given the text is is this the sum of X. I divided by N. Okay. Oh that my, excuse me. So here this little X. I just means it's a, you know, do anything with the I just it just means the things in the order. It's like an ordered list. So all this is saying X one plus X two plus X three plus yada yada yada plus X. X. And and you take that whole list of end things and divided by an which is what you've been doing for many years in math. But this is just the wave saying it in math notation. So let's take all these and together and get to mean so 76 plus 60 Plus 60 plus 81. Uh went in the room. Okay. Uh huh. Yeah, it's me. Make sure I've got them all 76, 60 81, 70-8080 68 and 73. Great. Take all that that they end in this case. There are nine 59 things. And that's going to give us a mean of Lead at 72.2 repeating. Yeah. All right. So there we go. That's the population mean. And population means just saying this is everything. This is everybody who's ever been measured in this this universe of of Of these nine students at this uh Sullivan's introductory statistics course. All right. So then part be determined to simple random samples of size three and compete the sample mean pulse of each sample. So from this population we're gonna sample three of them. I'm gonna pick this one on purpose. Uh I'm gonna pick Megan Jeff and Cami on purpose because that's also going to tie into our work here number three around part C. So it's 60 plus 60 plus 68 all the three. And we add these up and divide by three. We end up with 62.6 repeating. Okay, So 62 and 2/3 at your pulse rate. But this have the reason I chose, because that's substantially lower, That's arguably substantially lower than this pulse rate, than the population means two simple size. There's one let's do the next one. And this one again, I'm gonna kinda pick these out on purpose. I'm gonna do Jeanette kevin and Clarisse. Uh So it's 81 Plus 80 plus 80 older than three. Uh huh. And that ends up being 83 or 80.3 repeating 18. A third pulse rates. There you go. So those are two other sample means uh And the reason I chose those and I said I chose this for a reason is because that answers part C. As well. Which samples adult. And the sample mean that overestimates the population mean, in which samples result in the sample? Mean that underestimates the population means. So this one here, This underestimates the mean. Because look here's the means population mean. But this is lower. This is if you're using this in your evidence you'd say boy look look at all the pulse rates of these people. It's really low. But if you take the whole population it's about 10 beats higher. Same thing here. This 80 and a third pulse rate heartbeats are beats per minute is above the mean. So this overestimates the mean. And so it says which samples all in a sample means that overestimate which samples. So for parts, see anything that's gonna have your your values Less than 72, for example, I sort of 60, 60 and 60. Those are less than 72. So those are going to bring the mean down and then which samples result in a sample means underestimate sample. Mean that underestimates which samples? Uh I kind of switch the order of those. It was supposed to overestimate first. So anything that's over the machine will overestimate it. So like you're 80 80 81 and some combination of them, if you did, if you did 70-80 and 80, that also would overestimate the mean and then do any. So there's quite a few. So, anything that any, a lot of combinations here that could possibly overestimate the mean, 80 80 72. That's one set. We'll be another set. We did this one. This one already overestimates it changes. What else would well, 72. So this 73 In the 80s, that would do it too. Like even if you did 73, 80 and 72, that would also overestimated because that 80 would bring the mean up. So these are over estimators. Some under estimators. Mhm. Anything that would think this 72.2 repeating, that's going to be like your your your marks. So your values need to be less than that. So. So right here yeah That would go under 76, 80 eighties. That would Oh I know that would sorry where was that wrong? 60s. Look at that. That would bring your mean below the 72 mark Because the 60s are just like a big weight bringing. They're bringing that mean down. Uh let's explore this one then. Do any samples lead to a sample mean? That equals the population mean? So this is the population mean, 72.2 repeating. So this this is where it gets, it gets a little fun because we need to look at this process as a whole. So we want to figure out The sample size of three that also gives us this mean. And listening to how we got that mean, Which are all of these values, added them up and divided by nine. So that ended up being 650. So this is the population mean? How do we get it? We did 650 divided by nine. The sum of all the pulse rates was 650. The number of them was nine. And we are saying we want to sample mean? And our samples are all size three. We want some sample mean? Remember the sum divided by three. So let's just use some of that nice notation X I. Sigma X I divided by three to give us this exact value. Again this is one of the cool things and kind of get to understand what's happening here. So let's look at this summit. So we're looking for a sum. So a very specific some and hopefully we could pick it out from our population here such that this ratio would hold. And so let's uh let's sort of trying to isolate this little sigma X. I. So we're gonna multiply by three on both sides because we're trying to isolate this. That's going to cancel that out. I'm gonna do a few steps here. This three divided by nine is going to be Uh you'll be left with a three in the denominator. So 650 then by three equals sigma X. I. And so this this would be our son that we're looking for. This, this thing in red and I'm pointed you in red I should say is there as some here that we should be looking for off of our population And 650 divided by three. Get ready for our people. 216.6 repeating and we want that to be sigma X I The sum of three values and this has this .6 repeating this 2/3. And if you go back and look at our population there is nothing will clean up this work here. There is nothing That has a pulse rate of 2/3. There's no sample with a 2/3 pulse rate in it. These are all integer values. But this is saying the some in order for the sample of size three to give us this exact population mean you would have to have a some a sigma x i 216.6, repeating. And there's no way you can get that two thirds because there's no two thirds. That's pretty cool.

In this question, it was given that 0.8 export with equal to 74.0 and Standard Division with equal to 12.5. The pulse rate is 36 para minutes. To determine the difference between this pulse rate and the mean off 74 we get X minus export equal 36 minus 74 for equal to negative 38.0, moving to the next point B Given export equal 24 and stellar division also equal to well 0.5, the pulse rate will equal toe 36 31 minutes. To determine the difference. We get the difference all also equal to X minus X ball Negative 38.0, we divide these difference by the standard division yet X minus export over standard division with equal to negative 38 over 12.5 or equal to ST negative 3.44 Then this means that this difference, then this means that the difference is three point off forced on the divisions, moving to the next point which is C for the same givens. After we did your mind, the difference for equal to Nick to 38 going to all, and we divided by the standard divisions. We get negative 3.44 and we consider that there's the pulse rate off 36 ft per minute. Is three point offer standard divisions below the mean? Because it's negative. We measures that school, which is the number off standard divisions, that the value is from the mean and a positive is that the school indicates that the value above them and negative city school in the case that the value below the mean thus is the score is in this case is nicked a 3.44 which is a very equal to make the 3.44 which is below the mean moving to the next point, which is deep for your scene. Givens, after we get that school for negative three point off, all is that at the school is significant, and the post rate off 36 beats per minute is also significant and doesn't find an answer for this question. Thank you


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