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11.23-TAssume that the differences are normally distributed. Complete parts (a) through (d) below: Observation 49.2 45.3 43.5 42.9 49.6 43.6 45.4 52.4 52.9 46.4 45....

Question

11.23-TAssume that the differences are normally distributed. Complete parts (a) through (d) below: Observation 49.2 45.3 43.5 42.9 49.6 43.6 45.4 52.4 52.9 46.4 45.1 46.8 51.9 47.2 48.3 54.3(a) Determine d = X ~ Yi for each pair of data.Observation-1.63.9 -2.33.6-2.9 -1.9(Type integers or decimals ) (6) Compute & and s4:.2.625 (Round to Inree decima places as needed ) (Round to three decimal places as noeded; )

11.23-T Assume that the differences are normally distributed. Complete parts (a) through (d) below: Observation 49.2 45.3 43.5 42.9 49.6 43.6 45.4 52.4 52.9 46.4 45.1 46.8 51.9 47.2 48.3 54.3 (a) Determine d = X ~ Yi for each pair of data. Observation -1.6 3.9 -2.3 3.6 -2.9 -1.9 (Type integers or decimals ) (6) Compute & and s4:. 2.625 (Round to Inree decima places as needed ) (Round to three decimal places as noeded; )



Answers

Find the variance and standard deviation of each set of data to the nearest tenth.
{234, 345, 123, 368, 279, 876, 456, 235, 333, 444}

All right. So for number four, we are wanting to calculate the variance insanity aviation. So were given a list of numbers and that's the entire list of numbers that exists that's considered a population. So the first thing first you're gonna goto desk most dot com and you're gonna click, Start graphing and you'll get this screen. It'll be blank here, though. I type season. So Step two is gonna be in order. Find the variance. You wanna type V A r variants and then p for population, and then we're gonna start listing your number. So I started doing that right here. And so I'm going to type up those numbers. That's the hardest part of this. Is hyping up the numbers properly with commas. Now this one wasn't too bad because it's 3 21 3 22 3 23 3 24 3 25 3 26 3 27 3 28 3 29 a 330 Zehr last number. And so we have an exact answer of 8.25 is our variance for our entire list of the population. And then we can do one of two things weaken square root this 8.25 to get the center deviation. Or you can fine center deviation by typing Syrian STD for standard and then D e V for deviation p for population. Enlist those numbers now you don't have to necessarily re type battle those numbers. There is the lovely copy and paste option. So you'll do control, See And then down here in between these parentheses. Now, parentheses are important. Make sure that they're there if you forget the parentheses and they won't show up. So control v paste that list and there is our, ah, standard deviation. We're going around in the nearest tents. So our standard deviation is 2.9. And like I said, you could square root 8.25 and I'll show you real quick. And as you can see, that will get you the same answer

Okay. So for number four, we need to find the variance and standard deviation of the given list. And so the list goes from 3 21 goes in order, um, up until 3 30 So by accounting that out, that is 10 numbers. So the first thing we need to do whatever you're finding very interested in aviation is to find the meat the X bar. So we're going to make sure that whenever we add up all those numbers that we divide by 10 because there's 10 numbers in that list, So we're gonna go ahead and add up. 3 21 plus 3 22 plus 3 23 plus 3 24 plus 3 25 plus 3 26 plus 3 27 plus 3 28 plus 3 29 plus 3 30 And we get 3000 255. Okay. And then when you divide by 10 that's gonna get us 3 25.5 And so we definitely are. Could be using that average quite often. And step two because the pub Aaron's is your turn to see how first put out your mean is away from the numbers in the list. So I'm going to go ahead and we're going to be taking all the numbers in our list. And we do each of them subtracted by the mean. So I'm gonna be rewriting three or 5.5 pretty often. Here, it's clean. There we go. Sometimes the stylist has its own mind, I swear. Ah, it's a 3 25.53 25.5 the 5.5. And this is a great way to help keep yourself organized the way that this is kind of set up. It's a great way to show your work, because I imagine that is something you might have to be doing. Obviously, if you are allowed used technology and you were taught to use the technology methods, then feel free to do that as well. Just to build a check. Your word. Nothing wrong with verifying about the right answer. Okay. All right. So we're gonna be subtracting. I'm gonna be swearing now. The reason why we square is because sometimes you're gonna end up with a negative difference, and we want to always keep very insistent deviation positive. So by squaring it that will all ultimately always fix any negative. You might get somewhere, do 3 21 minus 3 25.5 And I get negative. 4.5. We're gonna square that. And that's going to get me 20.25 20.25 down. And then we're gonna do 3 20 to minus 3 25.5 That's gonna get us negative. 3.5 square that we get 12.25 3 23 minus 3 25.5 is 2.5 negative than you. Square that integrate at 6.25 Okay, so there, 24 minus 3 25 15 is in. The 1.5 squared gets, We get 2.25 and then we're gonna have 0.5 getting squared. That's me. 0.25 Right here. So then it pretty much because of how everything's going. If I do 3 26 minus 3 25.5 that's going to 0.5, squared 2nd 0.25 then this one is going to be two Point are 1.5 squared. That's gonna get me the 2.25 So you could kind of see the pattern because it's building back up on the positive sides. So we're gonna get 6.25 here, we're gonna get 12 0.25 and we're going to get 20.25 and then And lastly, to get the variation is we take all the numbers that are now in my box, all 10 of them. I'm gonna add them up and abide to get the average, and that average ends up being our variance. So we're going to do 20.25 plus 12.25 plus 6.25 plus 2.25 plus 0.25 plus 0.25 plus 2.256 45 plus 25 close 20.25 Add it all up. Get this 82.5. So 82.5 divided by 10 is going to be our variance and so are variants. Ends up Equalling eight 0.25 Gonna go ahead and put a circle around that and of course, we need to get the standard deviation as well. But luckily for us, that's not a very big or tedious step. We're gonna take whatever we just got in Step two, as are various 8.25 And we're gonna swear route the 8.25 and we're going to round to the nearest 10. So when we do that, you should be getting approximately you point nine as your standard deviation. Circle it and we're done with number four.

In problem number 70 were going to use a computer or calculator. I'm going to show you how to use the T I graphing calculator, the Texas Instruments, and we need to find the probability that one randomly selected value of X from a normal distribution with a mean of 584.2 and a standard deviation of 37.3 will have a value that is less than 525 for part A. This is part A. So when it comes time to use your calculator, um, on the graphing calculator and I'll pull my graphing calculator in, you're going to hit the second key. We're gonna hit the second key, and then you're going to hit the There's Key. Something hit the Bears key against second key than the variables cake, and then we're going to use number two, and that's your normal C D. F. So it's a cumulative density function. So when we use normal CD F notice below it, it talks about lower bound upper bound, so we're going to bring it to her home screen. So we're going to have our lower boundary. We're going to have our upper boundary. We're then going to have our average and our standard deviation. So in this particular problem are lower bound is infinitely into the left tail. So if I think about what the bell would look like, here's 584.2. We want to be less than 525. So we're going all the way into this tale. So in order to do that, our left bound is going to have to be a really big negative number. So what we're going to type in is we're gonna type in negative one times 10 to the 99 which is a super, super super negative number. So we'll have negative one times 10 raised to the 99th Power. Then we're going to do a comma. We're gonna do our upper boundary. So what we have just put in is a number all the way back here, so our upper boundary is going to be the 525. So we're going to type in 5 to 5, followed up with a comma. Then we have to go with our average, and our average was the 5 84.2. So we'll type in the 5 84 0.2 and we'll follow it up with our standard deviation. And our standard deviation was 37.3. So we're getting an answer for our problem. The probability that X is less than 525 is 5250.5 six to they took that away. I'm going to finish up what was hidden there. Remember that last argument for the A cumulative density function is going to be that standard deviation. Okay, so for part A, what's the probability that a randomly selected X value is less than 525 would be 5250.562? And we used our graphing calculator. And again I used the T I graphing calculator, the 84 or the 83 wood. We're all right. So let's go to part B in part B. You are asked to find the probability that X lies between 5 25 and 5 90. So the probability that X is between 525 and 590 again we're going to use our normal probability normal CDF, which means cumulative density function. And remember, it goes lower bound, upper bound average standard deviation, so we'll use normal CDF in this case are lower bound is the 5 25. Our upper bound is the 590 the average was the 5 84.2 and the standard deviation was 37.3. Some have slide my calculator in, and I'm going to do the second. There's select number two. This time lower is 5 25 separated by the comma, 5 90 separated by the comma, 5. 84.2, separate by the comma and 37.3. So for the answer here we get 0.5055 and I may have to move my calculator. All right, so let's go to port. See, Part C is a probability of at least 5 90. Probability will at least 5 90 means it's going to be greater than 5 90. So if you think about what the Bell would look like, then we're talking about going into this tale so our upper boundary is gonna have to be a super large number. So we're going to set it up by saying normal CDF again. It's lower boundary, upper boundary average and standard deviation, so we're gonna dio a lower boundary of 5. 90 because that's gonna be right here the beginning of our shaded region. Our upper boundary has to be a very large number. So we're going to say, one times 10 to the 99th power, our average was 584.2 and our standard deviation was 37.3. So I'm gonna slide my calculator in. We're going to set that up. Or did you? Second, there's number two. Our lower boundary is 5 90 separate by comma one times 10 to the 99th power separated by a comma put in our average, which was 5 84.2 and follow it up with our standard deviation of 37.3. So in this case or probability that X is greater than 5. 90 is going to be 0.4 three eight to So now that we have used the graphing calculator parte de wants us Teoh, verify these three results by going back to the old fashioned way and using the table three from the back of your book. So let's go back to we're going to party again the old fashioned way and part A Waas. What's the probability that X was less than 5 25? So the formula way is going to be to construct that bell curve, put the average in the seven or 5 84.2, we want less than 5. 25. So we will need a Z score. Remember our Z score formula this X minus mu divide by sigma. So we're going to find the Z score to be 5 25 minus 5 84.2 divided by our standard deviation of 37.3. So our Z score turns out to be negative 1.59 So if we're talking less than 5. 25 were also talking that we are less than negative 1.59 as a Z score. So we would then go to our standard normal table in the back of our book and we would get a 0.559 We're gonna also, um, do Port e as well at the same time. So are part a answer. If we sneak back to it, we had point 0562 0.562 So that's pretty close. All right. So part A, we got the 0.56 to, but when we redid it using the table of values, we got 0.559 So they're close. They both around 2.56 But the reason for the difference And this is the part e the explain the differences. The reason is, when we found this Z score negative 1.59 we rounded, um, the calculator. If you're doing it in the calculator, it's not rounding an easy scores. It's using more than two decimal places. So actually, the calculator is going to be a little bit more accurate because of the fact that we had to round this Z score. So let's do a comparison for Part B as well. So in part B, we had an answer. Let's go back and get it of 0.5055 But now we've got to do it with the long hand, using the old fashion way using tables. So Part B, the question was asking you what was the probability that we were between 525 and 590 so on our bell shaped curve, we want to be between 5 25 and 590 you're going to have to calculate the Z scores for each of those. So the Z score for 5 25 will be 5 25 minus the average, which was 5. 84.2. Divide by the standard deviation of 37.3 and you get a Z score of negative 1.59 and we're gonna find the Z score for 5 90. So we'll do 590 minus 5 84.2, divided by the 37.3 and you're going to get a Z score of 0.16 So when we're talking about being between 5 25 and 5 90 it's no different than saying between negative 1.59 is less than Z, which is less than 0.16 So we would find the probability that Z is less than 0.16 in our standard normal table. And then we would subtract from it the probability that Z was less than negative 1.59 Utilizing the table, we're going to get an area of 0.5636 minus 0.5 59 for an answer of 0.5077 And again, we're close, not quite exact. And again, the reason we're not exact is because over here we rounded to two decimal places. So we rounded within the problem, which skews our results a little bit. And then finally, for part C, let's do a verification on the part C. Move up a little bit, Part C. If we scooped back Part C was 0.4382 we had point for 382 and the problem we were solving was the probability that our value was greater than 5 90. So our bell curve 5. 90 would have been over here. So we need a Z score so it would have been 5 90 minus 584.2, divided by 37.3. And that's going to get us a Z score of 0.16 So when we were talking about being greater than 5 90 it's no different than Z being greater than 900.16 And since the standard normal table always goes into the left tail, we're going to have to rewrite this as one minus the probability that Z is less than 10.16 When we look in the table, we will get a value of 0.5636 resulting in a probability of 0.4364 And again, here we are. We're close. We had a calculator answer, and we have a manual table. Answer there close. And the reason there's a difference is because back here we rounded to two decimal places.

All right, remember, Six We need to find the variance in the Sudan deviation. So our first step whenever waiting to do that is to find the mean or the average we call that expert for short of the lists of given number. So this list has seven numbers, so we're going to be adding up all those numbers and dividing by seven. So I'm gonna take a calculator and add up the 400 plus 300 plus 3 25 plus 2 75 plus for 25 plus 3 75 plus 3 50 and we get 24 50 and rendered by that baseman, and we get exactly 350 as our average. So the average is very important because it gets used a lot in the second step. So our second step is to actually go ahead and find the variance. The variance is found by subtracting are mean from all of the numbers in the list. So they're seven overs and list. We're going to subtract all the numbers minus our 3 50 and then we need a square it because we're going to make sure we keep our various inter segregation positive. So you do 400 minus 3 50 and so on and so forth. So our again, our mean gets used a lot in a second step of the problem, which is OK, so we're just gonna be a lot of subtracting and squaring at this point. So 400 minus 3 50 is 50 50 squared is 2500 300. Ministry 50 is also technically 50 squared. So that's also 2500. Who ever So that look nicer. Great. One e 500. Ah, 3 25 minus 3 50 is 25. So we're 25 squared and we get 6 25 okay to 75 minus 3 50 is going to be negative. 75. But of course, when we swear we've become positive. So it's gonna be 56 25 4 25 minus 3 50 is 75. So if I've squared again that we just did this baby, that 56 25 again. Okay, 3 75 minus 3 50 is 25. Whenever I swear it is 6 25 and then there shooting mostly 50. That's an easy one. That zero in zero spurt zero So our last up to five variants is to add up all the person our boxes and find the mean of and that will be our variance. I'm gonna take 2500 plus 2500 plus 6 25 plus 56 25 36 25 plus 6 25 And we get 17,500 and then we divide by seven because there were seven items in the list and are mean or variance, rather is going to be 2500. So most the time you get a nice number here. Sometimes you have to brown there's tens of necessary, um, and then to get senator vacation because that was also part of the directions. You have to square root that now likely for us, the square root of 2500 ends up being a nice, easy number. It ends up being exactly 50. Sometimes you get a decimal most the time you get it s more other, and he'd have to round. But this time it's a nice hole number. Answer for number six


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