So in this problem, we're working with data from 50 Eminem's and they were weighed. And you need to start by finding the sample standard deviation. So we're trying to find s sub X. So I took it upon myself to put it into my calculator. I went into this stat feature of the calculator edit. And as you can see, I have the 50 pieces of data already in the calculator. So in order to find the standard deviation, we're going to hit Stat, we're gonna move over to the calculate menu and the one variable statistics. We're going to tell the calculator that we stored all of our data into list one. So in doing so, we find that our sample standard deviation is approximately 0.36 Part B. Part B is asking you to decide from a previous exercise whether the empirical rule is appropriate here. So you might not have access to that previous exercise right off the bat. But I'm gonna go back to my calculator and I'm going to look at the graph of it. So I have already put it in, and I've told it to give me a hist a gram. And when I hit graph, this is the shape you're getting, and that shape is what we would refer to as its bell shaped. So we could say, based on the history Graham from a previous exercise, it would be appropriate to use the empirical rule because the data is bell shaped. Another thing we could say is the distribution of data is approximately normal, right? So now let's go to Part C, which is asking us to actually use that empirical rule. Now, before we can use it, I think we should probably review what it says. So we're going to draw a bell shaped curve. And when you draw that bell shaped curve, we always put the average in the center, and then we count by standard deviations in each direction. And if we count one standard deviation in each direction, so that be that we're at Mu plus sigma and mu minus sigma, then you can be pretty assured that approximately 68% of your data is going to fall in there. So because the bell is symmetric in nature, if we take that 68 divide it in half, then each of these pieces would account for 34% of the data, and you can display it as either 34% or as its decimal equivalent. Now if we go two standard deviations out in each direction, so that means we're out mu plus two Sigma and U minus two Sigma. Then you could be confident that 95% of the data falls between these two values. So if you take away the 68% we've already accounted for, that means there's 27% left over meaning then that there's 13.5% in this little portion and 13.5% in this little proportion. They're a little portion then, if we go out one mawr standard deviation. So if we go out to mu plus three Sigma and Mu minus three sigma, then you are almost guaranteed to have 99.7% of the data to fall between those two numbers. And again, if you take that 99.7 and you subtract the 95% that we have already accounted for, that's going to leave you with 4.7%. And when I split that evenly into two pieces. Then this section here is going to have 20.235 And this section right here would be 0.235 And keep in mind, the entire bell is 100%. So there are little pieces beyond here that's still available. Um, and the entire bell would be that 100%. So now that we have reviewed that, let's look at the bell curve in terms of the data that we have. So we have the bell shaped data and we saw that by using the graphing calculator and in the center, we're going to put the average. And in the hint in this problem, they've already told us that the average was 0.875 And in the first step of this problem, we found that the standard deviation and actually the standard deviation should be an s rather than a sigma was 0.36 So if I add 0.362 that average, I will be at 0.911 And if I add 0.36 again, I will be at 0.947 And if I go back to that average and I subtract 0.36 I would be at 0.839 and if I subtract again, I would be at 0.803 So, in essence, I am asking you what part of the curve or what percentage of the curve would fall in between those two values. So if I look back, I'm going to standard deviations to the left and two standard deviations to the right. So that means I would expect 95% of the Eminem's in our sample to fall between these two values and the two values again, our 803 grams and 0.947 g. All right, let's talk about Part D. Part D is asking you what is the actual percentage of Eminem's that way between those two values? So whenever we're trying to find a percentage, the the easiest thing to do is to think in terms of parts and whole, and when it comes time to the whole in percentages, that would be 100% When it comes time to talking about the whole with regard to our data, all 50 would be the whole data set, and we've got to figure out how many of those 50 are between these two values. So if we go back to our calculator and I get back into my data set, I've got them in list one, but they're not in order. So it's not really easy to tell how many are between those two values. So I'm going to go back to stat, and I'm going to tell it to sort the list. So we're going to sort list one so that I could now look at the list and I can see that 0.79 is not in between 0.803 and 0.947 But then when I get to the second data point, it is. And as I keep scrolling down, the last one is not in this range. So really, there's two pieces of data that do not fall in this range, meaning that there are 48 pieces of data that do, and I need to know what part of the 100% that would represent. So if I were to cross multiply here, I would find out that the actual percentage is 96%. So the empirical rule told us 95%. Our actual data is 96%. Let's look at part E. Part E is asking us to use the empirical rule again, but this time to tell us how much is in between the values of 0.803 No, sorry is more than 0.911 g, so we want more than 0.911 So I'm going to reproduce my bell and again. At the center was 0.875 And when we went one standard deviation over, we got that 10.911 When we went to standard deviations over, we were at 0.947 and when we went to the left, we were at 0.839 and to the left one more time. Put us that 10.803 and this time we're trying to determine the percentage based on the empirical rule that would be greater than 0.911 Keeping in mind that the whole bell is 100% we're going to take away half the bell because half the bell does not meet the criteria. So we'll deduct that 50% and then we're going to take away this little section here, which accounted for 34%. So the part that is shaded the part that is greater than 0.911 is going to end up being 16% of the data should weigh more than 0.911 Gramps. And for part F, it wants to know what's the actual percentage of Eminem's that way more than the 0.911 So we're going to do like we did in part D. So for part F, we're going again. Think in terms of parts and whole in terms of percentage, the whole is 100. In terms of our data, 50 pieces of data would be the whole set of data. We're trying to figure out what part of the 100 and we're now going to look at our data, and we're going to see how many pieces of data arm or than this 0.911 So again, I'm gonna bring in my data that's already been sorted and bigger than 911 is going to be the 0.9 to the 0.9 threes. The 0.94 point 95 So we're gonna have 123456 pieces of data so we could put a six in here. And if we were to cross multiply 50 X equals 600. So therefore our actual is going to be 12, so we could say 12% of the data is actually mawr than the 0.911 g. So the empirical rule told us that 16% of the data should weigh more than and we came up with our actual data to be 12%. So, as you can see, the empirical rule is a framework. It's not perfect. It's giving you an approximate percentage off values in between certain locations on that bell curve.