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E(X - %)2 S = n -1What is the variance for both players? Based on what we have talked about in class, what does the SD tell you about the spread of scores in the di...

Question

E(X - %)2 S = n -1What is the variance for both players? Based on what we have talked about in class, what does the SD tell you about the spread of scores in the distribution for each quarterback? In any given year, which quarterback can you count on to consistently throw passes close to his average; McNabb or Manning? (Note: This isn't asking which player is better, it's asking which player is most consistent with their average)_ What do you think would happen to the standard deviatio

E(X - %)2 S = n -1 What is the variance for both players? Based on what we have talked about in class, what does the SD tell you about the spread of scores in the distribution for each quarterback? In any given year, which quarterback can you count on to consistently throw passes close to his average; McNabb or Manning? (Note: This isn't asking which player is better, it's asking which player is most consistent with their average)_ What do you think would happen to the standard deviation if you removed the lowest value from McNabb's stats? (Think back to our lecturel)



Answers

This histogram displays NFL data from the 2010 football season reporting the number of yards gained by each of the league’s 191 wide receivers:
The mean is 397.15 yards, with a standard deviation of 362.4 yards.
a) According to the Normal model, what percent of receivers would you expect to gain more yards than 2 standard deviations above the mean number of yards?
b) For these data, what does that mean?
c) Explain the problem in using a Normal model here.

So in this question, we are given a bunch of data for the season winds for football and were asked to calculate the range mean variance and standard deviation for this population data set. So since this is a population, basically we have our I mean variance, standard deviation and range. So for range, we basically take the max value minus the mean value. So we have 13, which is the Max and our men, which is to so range is 11 now for the mean new. We basically have the some off all the values divided by the number of values. So we have 13, 10, 12, 11, 78 plus six plus six, 10, 7, 12 all the way up to two. So we're basically taking all of these values and we're dividing it by the number of values which basically 123 45678 times to that 16, which gives us value off 7.56 to 5. So that's our mean right here now for the variants that's basically use our formula. So we have we have all of the values. We basically have to calculate the value minus the mean than the value minus the mean squared, after which we take our some off X minus mu square, divided by 10. So that will give us the variants and then the square root Uh huh. This value will give us the standard deviation. So let's just see what this looks like for a few values. And then we can put that into the calculator. So let's just take our first three value 13, 10 and 12. So we have 13, we have 10. We have Well, I'm gonna do 13, minus 7.56 to 5 gives us 5.4375 square off. That is 29.566 when we do the same thing. So we have 10, you have 10 minus 7.56 to 5 gives us 2.4375 square off. That is 5.941 now 12 minus 7.56 to 5 will give us 4.4375 and then the square off that it's 19.691 So we're basically gonna do the same thing for all of the values. And after doing that, we're going to some this column up and put that as our numerator. So let's go ahead and do that. Doing all of that basically gives us our numerator to be 181.9375 and our denominator number off 16. And so that gives us the variants. That's 11.37 winds squared, so that's our variants. And our standard deviation is basically the square root off our variants, which gives us 3.37 to 1 prince. And so this is thes are the answers to the question. This is the variance, the standard deviation, the mean and the rich.

Ready up, we're going to be looking at the Super Bowl football game and the new Orleans Saints, how did they win it? Well, this question thinks it has to do with the numbers on their jerseys. So we're gonna go ahead and calculate the range variance and standard deviation of that. So what we can do for our range is simply find the largest uh the biggest number we can on their roster And that is going to be 88. Yeah, Yeah, bonus points, if you know who that is, And we can find our low value, which is going to be number nine And that gets us a range of 79 uh huh. And there aren't really any associated units with this because they're just jersey numbers, so are variants, we're going to calculate that by taking the difference between the mean and each of these numbers that we have in our roster, squaring that difference and then adding those all up and then finally we'll divide that by the number of observations we have. So why don't we calculate that mean? The mean is going to be the sum of all of our numbers, Which is going to be 554 And then we'll divide that by a number of observations. So there's 11 of them, of course the football team is bigger, but I think we're just looking at the winning line up and this gets us an average of 50.364 Mhm. So we're gonna take that number and plug it into our variance formula. And what we're gonna do is some up each of our individual terms, the excise, the rosters Minour that average of 50.364, we're going to take that difference and square it And divide by the number of observations we have -1. Mhm And then this is going to come out to be a variance of 1017 0.65 and they aren't really gonna be units for this, but we can calculate our standard deviation by simply taking the square root of that. So if you get the standard deviation, we're gonna square both sides and that means that we will have our 1017 B squared 65, so we're gonna square that to get our standard deviation, I shall we can take the square root, blah blah blah. Okay, yeah, so we have a variance here, that's the square of the standard deviation, So if we take the square root of our variance, Mhm That's gonna be 1017 0.65. Yeah, And this is going to come out to be a standard deviation of 31.9. And just for context, it doesn't really make sense to find the standard deviation of jersey numbers and start basically names, you might as well be averaging letters and it looks like we've got a team that one who had a lot of a's in their names or something like that doesn't make any sense and probably should look at something, like how well they can throw, how well there's a quarterback through how many times they got sacked, that kind of stuff. You don't really hear the sports commentators making comments about, oh, they've got a really good jersey lineup.

Alright, So we're even with the data set were asked for the range Standard Division in various. Okay, So the easiest value that can obtain is the rich just equal to the difference between the maximum value and the minimum value. So look at the data set. We can see that the maximum values equal to 99 while the minimum values to serve it. Okay. Hence the range is equal to 92. Mhm. Now, in order to calculate the standard division, we have to get the mean first. Okay. So mean is equal to the summation off each data set. Okay. Divided by the total number of data set. So we have the girl, we're going to add all the data in the data set. So we have 89 plus 91 plus and so on, so forth until we reach. Oops, mhm and so on and so forth until we reach 82 plus 60. Okay, we sum them up and divide them by total number off. Uh, data in the data sets. We have 11. Okay. And should have a value off me using your calculator. That's equal to 57 0.1. Uh huh. now the reason why we go, Why we need to get the mean, uh, this issue noted as exporter can because the formula for the standard Division is equal to the square. It a mhm parrot off. Uh, the submission. That's all right. This one. We have the square root off the summation off, except by minus the mean squared divided by the total number of samples. A local number, all they'd set. Since we're given with sample here, this denominator will become an minus one. Okay, now X of I represents each value in the data set. So we have here for example, 89 minus the mean That's 57.1 subject and get the square plus the next value. Anything sets 91. Why, Sufi 7.1. Get the square and so on and so forth until we reach 60. By this 57.1. Get square and divide all of them by 11 minus one and close them in big spirits night. Okay? And the standard deviation that we should get is equal to 33 point line. Mhm. And to get the variance variance is just the square off the standard divisions and just square 33 point line. Okay. Should get a value that's equal to 1152 A point seven or point meat. So this are over.

All right guys. So what we're gonna do for these is we are going to find the population means the range and the population standard deviation for the eagles, offense and defense. So here we go. So the population mean here's how we're going to do that. The way we do that is we're gonna click equals on our keyboard and then average of all the offensive players. We highlight all the offensive players right there, typed in Those are typed in from the text there and our average is just a hair over £260 that I like to use three decimal places. Let's go and do that. And we do the same thing for the defense. And so what we'll do there is again equals average of all the defensive players. Well, it looks like the defensive players are just a touch lighter than the offensive players. They're averaging out to 45 the office players are average around 2 60. Okay. Now, for the range. Okay, So for the range, what I'm gonna do is this is a little bit more complicated but instead of having to sift through all the data and find the biggest one in the smallest one, here's what I'm gonna do and you can do this on google sheets or excel by the way. So on this google sheet I'm gonna equals max and then I'm going to select all the offense before I go on from theirs and then what I'm gonna do, okay, it's not working, hold on, Put equals the max. Sorry, I'm having a little bit of trouble here, highlight all of that. Perfect. Okay, and then okay, there we go. So yeah, okay, and then now I'm gonna do so that will find the maxim in there. It'll find the max for me and then I'm going to subtract the minimum. So it will also find the minimum for me, which is kind of nice. Well that Okay, cool, and so I'll find the biggest one and subtract smallest one and there we go, right there. Okay, awesome. And I can do the same thing over here, just highlight all the defensive players. Uh huh. And there we go. So the range, so it looks like, well the offense is a little bit heavier, they're also a little bit wider range today they also have some built a wider range of weights as well. All right, um and then, okay, so the population standard deviation, let's see what's going on there, and so I do equals S. T dev P. Right there, and I'm going to highlight all the offensive players and there we go. So its population standard deviation About 50 little over 53 and the same thing here. Population sp dev he is all the offensive players and there we go. Okay, so it looks like um so the offense has a higher mean, higher range and a higher standard deviation as well, and so which has more dispersion? So the offensive players are more dispersion. Why not? Because they have a higher mean? By the way, the higher mean does not tell you about dispersion, It's what tells you about dispersion, our range and standard deviation. So because they have a higher range and standard deviation, the offensive players have more dispersion.


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