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Find the mean for the data in Exercise 2 from the grouped frequency distribution found in each of the following exercises.

a. Exercise 2

b. Exercise 4

a. $\overline{x}=\frac{\sum x f}{n}=\frac{5550}{80}=69.375$

b. $\overline{x}=\frac{\sum x f}{n}=\frac{5550}{80}=69.375$

Multivariable Optimization

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Johns Hopkins University

Campbell University

Harvey Mudd College

Idaho State University

this question gives you a data set and ask you to use frequency tables to find that mean the data set was given an exercise too. And we made frequency tables and exercises two and four. Part a wants us to use the frequency table we made in exercise too. This frequency table has had been with arranges of 10 starting at 30 we had ranges 30 to 39 40 to 49 50 to 59 60 to 69 70 to 79 82 89 and 90 to 99. And the frequencies we found where 16 13 22 17 13 and eight. Now we know when we're using frequency tables to find a mean we have x bar are mean is the sum of X times f all over n where n is also can also be found to be the son of F So we know we have our frequency. That's this column here. That's their f column. But what is our ex call? What are our values? We can't really use this range of values because, Well, how would we do that? So, to reconcile this, we take X to be the midpoint of each of these ranges. I'm gonna write it. Make a new column called X with the midpoint of each of these ranges, which is 34.9, a 44.9 and so on 54.9 what we'll do next. Excusing that 95 34.5 44.5 from just two races. 5555 And then we'll have 74.5 84.5 94.5. What we'll do next is have another column X times f where we just multiply these values times their frequencies to get what we need for the numerator. When we do this will get a serious numbers. 34.5 in the first column, 267 then 708.5 141,419 1266.5, 1098.5 and 756. Now we have all of our F values. Oliver XF values. What we need to do now is some them up. I'm gonna make it another road down here for the sums. So when we sum up all of our frequencies, we're going to get 80. That makes sense because we had 80 data points to begin with. This is what we'd expect, and that's a good check to see if you're off in your frequency table. If you have a different number of frequencies, when we some of our X times F values, we get 5550. So now all that's left is to divide the two. We know X Bar is going to be the sum of X f, which is 5007 or 50 divided by n, which is some of F, which is 80 and we'll get that X bars equal to 69 0.375 So now let's do the same thing with the other frequency table that we that we made. Why do we do it twice? Well, what's the use here? Um, the reason we are checking it like this is because when we're using frequency tables to get a mean, it's only an approximation. Um, we don't have since we're not adding exactly the numbers that we are given, Um, we don't have the exact total of all the all of our data points. These are only estimations of the mean. They require fewer additions so that little bit easier. But they're a little bit less accurate and can be skewed a little bit. Say, if we had different ranges or arranges started in different places, things might change. So let's take a look at the second, uh, frequency table that we made this time. We so used ranges of length 10 but they started at 39. So we had 39 to 48. 49 to 58 59 to 68 69 to 78 uh, 79 to 88 89 to 98 is there are are ranges. We also had frequencies. This is from exercise for 6 13 20 1913 and nine. Now, again, I'm gonna do the same thing. When use the mid points for each of these are X values will have 43.5 53.5 and so on. And then I multiply each by the frequency so we'll have our x f column. Six times 43.5 is, uh, 261 to 61 will get 695.5 for our next one. Then 1270 then 13. 96.5, 1085.5 and 841.5. So now again, well, some of the f sent all the X efs All the F's will some to the same thing. We still have 80 data points, so and is 80 and then when we some are accepts this time we also get 5550. Now, this is just by coincidence that it was possible to get a different some of X efs. But in this case, we got the same thing when we took two different frequency tables. So now we need to do is divide. We know export is gonna be 5550 over 80 which again is 69.375 And there's your fine