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Find the mean for the data in Exercise 1 from the grouped frequency distribution found in each of the following exercises.

a. Exercise 1

b. Exercise 3

a. $\overline{x}=\frac{\sum x f}{n}=\frac{2657}{36} \approx 73.81$

b. $\overline{x}=\frac{\sum x f}{n}=\frac{2682}{36}=74.5$

Multivariable Optimization

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this question asks you to use a frequency table to find the mean of a set of data. Now the set comes from exercise one, and A Nectars has one and three. We created two different frequency tables. So part a of this question asked us to use the first frequency table that we made. Now, keep in mind when we're finding means from frequency tables like this, these air only estimations of the means the value that we get will depend on how we group the data and we might get different answers accordingly. The first frequency table used groups from 0 to 24 size of 25. It looked like this 0 to 24. 25 to 49 52 74 75 to 99 100 to 1 24 and 1 25 to 1 49 So those are our ranges. We had frequencies which will call F of four, eight, five, 10 five Earth exceeding four and five. Now we know that using a frequency table to find them mean we're gonna use is the equation. Export equals with some of X times F that is all of the values times their respective frequencies over the sample size and the sample size. And it's just gonna be this sum of all the efs. So we have the EFS here, but we don't actually have the exes. We have ranges here, but to find the X is the values. This method asks us to take a midpoint of the range, so I'm gonna go ahead and get the midpoint of all these ranges. Will have 12 37 62 87 112 and 1 37 Now we have X and F, but we need to find x times f. I'll make another column on our table and we'll put in the X Times EFS 12 times four is 48 37 times eight is 296 62 times five is 310 87 times 10 is 8 70 112 times four is 448 and 1 37 times five is 685. So now we have all our X times, efs and all of our efs. The only thing left to do now is to sum them up. I'm gonna create a some row, and we don't really care about the some of the values because they don't That doesn't represent all of our data. And we can't some up these ranges. But we do care about the some of the f con their frequencies when we add all these up. Or if we just count the number of data points in that table, we find that there are 36. And then when we sum up all of the X times efs, all of the data times their frequency, we get value of 2657. So now we know that the some of XM's F is to 657 and in, or the some of the EFS is equal to 36. So that means X bar are mean. He's gonna be 202,657 divided by 36 or 73 0.81 As just a estimation, dismal approximation. They're a lot more, uh, decimals. After that says one way of finding the mean using frequency table. But we also created a separate frequency table. Um, that we're gonna we can take a look at Let me scroll down here and I will. Right are other frequency table in this one. We use bin widths of 20. So we went from 0 to 19. 20 to 39. 40 to 59. 60 to 79. 80 to 99 100 to 1 19 1 22 um, 1 39 and 1 40 to 1 59 So I'm also gonna write down our mid points this time. Our mid points. Don't line up exactly how we like. We'll have a 0.5 in them. So are good points or 9.5. 29.5. 49.5. 69.5. 89.5, 109.5 129.5 in 149.5. Remember, those are X values Wilson into Ridenhour frequencies. We found these in problem in. Problem, actually, is three remember it? It is 45 for 59 three for two. Those are frequencies. Now, let's do the same thing. We did what we find x times f extents f here. Um, your first column is gonna be 38 then 147.5. Then 198 347.5 805.5 328.5 5 18 in 299. So, like we did with before, we will add a totals column at the bottom. Doesn't matter what the values their total to can't add up the ranges again, our frequencies will total up to 36. Makes sense. There were 36 data points to begin with. That can ever change, but now our, uh, our X times fl use some of to something a little bit different. Now, when you add them all up in a calculator, you get 2682 so a little bit different. And to find the mean this time will do 2682 divided by the total frequency 36 we'll get something different. We get 74.5. So again, these are just ways to estimate the means using, um, frequency tables, their estimates because they're not always gonna be perfect here. We estimated that it's that it's 74.5 above. We said 73.81 um the true mean is actually somewhere in the middle, I think, Uh, yeah, 73.86 So it's not perfect, but it gets you pretty close, and that's that is your final answer.