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Problem 182 Hard Difficulty

Each year, Sports Illustrated ranks the highest-earning athletes in the United States. Their earnings include their salary/winnings as well as endorsements. Often the endorsements are more lucrative than their earnings. The top 20 earnings for 2008 are listed in the following table (in millions of $\$$ ).
a. Find the mean earnings for the top 20 highest-paid athletes.
b. Find the median earnings for the top 20 highest-paid athletes.
c. Find the midrange of the earnings for the top 20 highest-paid athletes.
d. Write a discussion comparing the results from parts
$a, b,$ and $c$
e. Find the standard deviation of these earnings.
f. Find the percentage of data that is within 1 standard deviation of the mean.
g. Find the percentage of data that is within 2 standard deviations of the mean.
h. Based on these results, discuss why or why not you think that the data are normally distributed.

Answer

a. $35.9 Million; b. $28.5 Million; c. $75.5 Million; d. Midrange and Mean are relatively high because of one large salary; e. $23.5 Million; f. 90%; g. 95%

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Video Transcript

okay for this problem. We're giving some us information about salaries of athletes, and we have a lot of things we want to find here the mean in the media in the mid range. Well, first thing we need to do is kind of we have enough values you're gonna use are calculated a types of in. So let's just take these in. We'll talk about it as we go. Um, so we're gonna do the edit here. We're gonna just take our list numbers in way. Have 1 28 and these air millions of dollars. Have you, Pete, let's type of these in and think about this so you can see either It's, um a lot of spread of this data said. Now we got down to the four East in There's 35 35. Do a couple things here with the technology to help us answer some of the eventual question because not only do they want us to find the meaning and in some other information, but we're also gonna talk about standard deviation, percentiles and whatnot. So let's look at this to get around 35. 31. A lot of values here in the twenties and thirties. Now it's done here. What I do after a table. Numbers, animals gonna check. Make sure you have the right number of numbers. It is very easy to miss a number. Some, even by talking and typing in same time, I may miss one. So the good news is, it looks like I end up with $23 million for last entry, and it says we have 20 athletes, so looks like that 20. It looks like I have a least the right number of numbers If I didn't do any typos Now, that's something I do here. Ah, because it be nicer to see these numbers. Yes, they are kind of in reverse order. Right. So they go from highest to least. So I was going to sort him from least highest. But I know that I'm looking at the date. I don't need to re sort those. I thought they were not in any type of order. What I was going to say. I was going to use a source if they weren't in order to do this sort A, you make him in order. But actually, because the highest listed first in their somewhat in order. We're okay. So I'm gonna go right to my calculations. Step here. So one variable statistics is gonna answer most of the questions for us. 71 bar stats and my list One data. You know where the secret got here. Now look back at my question. That was asked. So they said, find the mean earnings of the top 20 highest paid athletes. Okay, so for part a, the mean is the x bar. So the mean salary is 35.5. Let's say 35. What? And this coming along video, Someone gets gonna work through that. And then we look at the second thing. The median salary. So didn't give us the median salary. So calculator doesn't give us that. I'm gonna go back and just look, we gotta find in the middle number. Backed a habit on the screen, So I now have 20 values. Um, So the middle number is 123456789 10. So, between 30 and 27 or the is the do the middle. So just like in see your ensure you what I did on my computer screen sounds like back here at the edit. So since there's 20 between the 10th and 11th I should say 123456789 between 30 and the 27. So between 30 and 27. So that's gonna be 28.5 million is the median. Okay, so then a 2nd 1 Um, the mid range will mid range is the athletic meet of the highest and the lowest. So since I have it on my screen here, let's look and see that the lowest WAAS 23 in the highest is 1 28 So let's take that. So the mid range, the mid range equals two 1 28 plus 23 divided by two. So let's see what that is. 1 20 1 51 divided by two is 75.5. Some throws us off a little bit as far as the average show because that one high salary drags us way up. So 75.5 million is be number for the mid range on the party. Let's see what we want to find here. Ready discussion. Comparing the results from Parts A, B and C. All right, so let's look a B and C So the middle number is a pretty good representation. That's, uh, yes, we're taking here. So I see that the because of the of the higher value mid range looks really high also mean IHS pretty high, because from the one large value, the median is probably the best represented them. So representative salary is like the 28.5 million medians by the best representation of what the crew salaries is a little bit bigger for you. So that thanks to this thought here, this way up a little bigger, well readable. Okay, um, keep going here. So we want to find the standard deviation of these earnings. So let's go back to our It's just is easy for me to calculate against was stopped. Calculate one of our stats. So from memory and what's gonna paste this in just because we have it here just in case. So standard deviation is, oh, 23.6 million, two or 3.5 million. I should say standard deviation in for part F by the percentage of Davis within one's dinner. Deviation of the mean. Okay, so let's look at our numbers within one standard deviation of the mean, so within one standard deviation of right plus or minus one SD equals. So we have actually discount and see how maney there. So one standard deviation. Uh, so then that means for mean ISS 35.9 look and plus or minus one standard deviation. That means we're gonna have 23.5 million. So you plus 23.5 million. So that was 14 589 59.4. So bits in within one sen innovations in less than 59.4. The difficulty plan here. Or we can say, in the low end, you didn't think of 35.9, minus 23.5. So this calculator here 35.9 minus 23 point five, 12.4. So, basically, how many numbers fall between 12.4 and 59 point for somebody? You there, old pal interpretation. So if we do, that means the number of numbers. Let's use the letter in number number is that air between 12.4 and 59.4? Well, let's look back at her and her date over here and it's kind of easier to look at. What? How many don't fall in that me in that range here so that it stopped? Yeah. So are there any above 59.4? Well, there's just these 2 62 and that one. So there's two that are above one standard deviation, and how many are below 12.4. Don't think there's any, so there isn't any so, but we want to know. So that means 18. So the only those two don't fall within that. All right, so let's look at the percentage of that means that there's 18 out of 24 within plus and minus one standard deviation. So that is 90% Okay, 20% fall within plus or minus one standard deviation. A little bit more room here. Now let's do this down. Fergie. So which, plus or minus two standard deviations? Well, two standard deviations. I'm not going to low, and so none fell below that I was going to be up brand. So let's just think of our mean, which waas 35.9 million, like it's just down one screen here. So let's see this. 35.9 million and plus two times only 3.5 million. I think I kind of know the answer to this already. But to be above two standard deviations means it has to be above that number. Let's look so he is in my hand calculator here, 35.9 plus two times 20 people. Five. That's any 2.9. So to be to be above that means it has to be greater than $82.9 million. So there's only one number that will one really rich person when really rich baseball player. It does that. So that means, um, 19 out of 20 fall within plus or minus two standard deviations. So that's 19 out of 20. So that means 95% fall within plus or minus two standard deviations. All right, that's a hardness question there. So, as you can see, the biggest deal is that that one big guy, it would be an out liar. No one got lower, kind of pulls all the data up, and he lives. Uh, what were two standard deviations above the mean and that makes him over once dinner deviation above the mean as well. So we've addressed all of our questions