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7102.16Quesion HepConsider the following dala trom two trcpendent samples wilh equal population variances Corshud 9974 conlaluce ponnlion (noan 5 Assuna th: populab...

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7102.16Quesion HepConsider the following dala trom two trcpendent samples wilh equal population variances Corshud 9974 conlaluce ponnlion (noan 5 Assuna th: populabon variances arc intewvall eclirale Ina: diflerenct cqual and Inalinne populabots are nolmaly distbbulcd 67 5 12 2Click huru Io_burthat Adishitxnlion Ialle. paqa 1 Clckhtollo seauha ( dsbiytbn lablo Dquiz 99* conldonce Ilorval # (DD (Round two decinial pLices 45 needed )

7102.16 Quesion Hep Consider the following dala trom two trcpendent samples wilh equal population variances Corshud 9974 conlaluce ponnlion (noan 5 Assuna th: populabon variances arc intewvall eclirale Ina: diflerenct cqual and Inalinne populabots are nolmaly distbbulcd 67 5 12 2 Click huru Io_burthat Adishitxnlion Ialle. paqa 1 Clckhtollo seauha ( dsbiytbn lablo Dquiz 99* conldonce Ilorval # (DD (Round two decinial pLices 45 needed )



Answers

Based on Data Set 1 "Body Data" in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4 (All units are 1000 cells/\muL.) Using the empirical rule, what is the approximate percentage of women with platelet counts a. within 2 standard deviations of the mean, or between 124.3 and $385.9 ?$ b. between 189.7 and $320.5 ?$

The question here basically gives us the fallen results for independent samples taken from two populations. And it wants us to do the following so hard a year. It wants us to find the difference between the two population proportions. So essentially, to find the difference, we could just take P one minus p two. And that is gonna be is your point for eight minus 0.36 which is your appoint 12 here for part B here asks us to develop a 90% kind of confidence interval for the difference between these two population proportions. Do such first of all we need to We know that the confidence level is going to be 90% and we know that to find the Z value for this So the A over to here, where a here is going to be 10%. So Z or Z values gonna basically be aligned with 0.5 So using the table found in this particular graph, we know that this particular value is gonna be 1.645 and thus the endpoints of this confidence interval here will essentially be the difference between these two proportions. So it zero point for eight minus 0.36 and then we subtracted with this particular Z Valley here to subtracted. But 1.645 and we put it through the end point formula, which is going to be the square root of P 11 minus P one over and one plus p two multiplied by one minus P two over and to here. And that is going to give us approximately, at least for the first interval here that is going to be 0.586 and the other interval. We just need to basically add 1.645 to this particular equation. And that's going to give us the other endpoints of these. Your a 0.1814 here. So for apart, See here it wants us to develop a 95% confidence interval. So in this case, we want to solve for the values a so one minus A has to be 0.95 so a is going to be 0.5 So we know that the Z a over to here is going to be Z 0.0 25 and which is going to correspond to a value of 1.96 so plugging into the same exact equation up here? Except we're just replacing this particular value here with 1.96 we're going to get the intervals of 0.469 to 0.19 one.

The first thing we have to compute is the differences between each element from each population. So 11 minus eight is equal to three seven minus eight is equal to negative. 19 minus six is equal to three. 12 minus seven is equal to five. 13 minus 10 is equal to 3 15 minus 15 is equal to 0 15 minus 14 is equal to one. And now we have to find this is the answer to part A. Now we're asked to find the bar which is just the mean of the differences so that is equal to each individual mean the sum of each individual need over the number. Sorry. The sum of each individual difference it over the number of differences which is equal to In this case, the sum of each of these individual data points over the number elements which is seven, which is equal to two. And now we have to find a sample standard deviation which is equal to the square root of the some of the difference between each individual difference and our mean difference squared over the number of differences minus one soldier to a new page. For this equal to the square root of the Somme. Each individual difference minus mean difference over a number of differences minus one which is equal to approximately 2.817 This is the answer to part C. And now we're asked to find a point estimate for the difference of population means. And this is so the difference of population means is also our deep are which is equal 22 And now we have to come up with a a confidence interval. So we're asked to find a 95% confidence interval. So to do this, we will use, um, the following formula. Our confidence interval equals D bar plus or minus a T to t statistic. Because we're not given a population ah, standard deviation. We compute the sample standard deviation a T statistic for Alfa over to where Alfa equals one minus the confidence level. So that is equal to one minus 10.95 equal 2.5 So de bar plus or minus our T statistic for half of our Alfa times wth east andr deviation, the difference is over our sample size. So in our situation, D bar is equal to two plus or minus. I'm just going to write t of 0.0 to 5 for now because Alfa over to is 0.5 over to which is your 0.0 to 5 times 2.817 over the square root of seven. In order to find our T statistic, we have to compute a degrees of freedom and our degrees of freedom is equal to end minus one, which is equal to seven minus one, which is equal to six. Now, using a tea table, we can find out where a significant Slobo of Sierra 0.25 associated with a degrees of freedom of six lies. And we get that we have a confidence interval of two plus or minus 2.447 times 2.817 over the square root of seven. And that means our final answer is our were 95% confident that the true average difference for our two populations lies between our lower end of our confidence. Interval 0.747 and our upper end of our conference interval 3.9253

So as we introduced normal distributions, there's something called the empirical rule which states that in our normal distribution, 68% of our data is gonna be within one standard deviation of the mean or the mean is centered right in the middle? All do not mean by this meal right here. So for red, that's going to be one standard deviation away from the mean. Which is going to hold about 68% of all our data. at two standard deviations away, We are going to encompass two standard deviations or 95% of all the data. Yeah, It might be helpful to draw these arrows right here. So that's going to encompass 95 and 68% is going to be within this At the 3rd standard deviation We're going to get 99.5% of our data. I didn't really make my alliance long enough, but you get the idea, Yeah, we're gonna have most of our data. All but half of a percent of our data is going to be within three standard deviations. Now, for part A it asks, what is the approximate percentage of women with platelet counts that are within two standard deviations of the mean? So that's gonna be two standard deviations or 98%. So there we go, that's gonna be 95%. Yeah. And then for part B Between 215 and 345, well now we have to interpret this. Okay, if our mean is 280 and our standard deviation is 65. What are our bounds? Well, let's go out one Let's start with 280 And add 65 to that. What do we end up getting? Well, it's gonna be 20 to get to 300 plus the 45 that are left, so that's going to make 345 for our upper bound, Let's ask, see what they're asking. 345. Okay, so reasonably 215 is going to be 65 away from 280. This means that we are one standard deviation From the mean, and therefore that is going to have 68% of our data, So that's gonna be 345 and 2 15, Okay.


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