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145. (5 points) Generate pairs (Ci, yi), 1 < i < 10 where Ti = 2 and yi = 1+ Zi, and the Zi are standard normal random variables: a) Verify that the slope of ...

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145. (5 points) Generate pairs (Ci, yi), 1 < i < 10 where Ti = 2 and yi = 1+ Zi, and the Zi are standard normal random variables: a) Verify that the slope of a linear model for y ~ x is close to 1 b) For each 2 , change the value Yi until the slope is < 0. This does not have to be done ecactly: A whole number value of the new yi is sufficuient: Is this always possible? What do you observe about the values that the new Yi must have? Can you explain this?

145. (5 points) Generate pairs (Ci, yi), 1 < i < 10 where Ti = 2 and yi = 1+ Zi, and the Zi are standard normal random variables: a) Verify that the slope of a linear model for y ~ x is close to 1 b) For each 2 , change the value Yi until the slope is < 0. This does not have to be done ecactly: A whole number value of the new yi is sufficuient: Is this always possible? What do you observe about the values that the new Yi must have? Can you explain this?



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A data set consists of ten $(x, y)$ pairs of numbers: $$ (3,20)(5,13)(6,9)(8,4)(11,0)(12,0)(14,1)(17,6)(18,9)(20,16) $$ a. Plot the data in a scatter diagram. b. Based on the plot, explain whether the relationship between $x$ and $y$ appears to be deterministic or to involve randomness. c. Based on the plot, explain whether the relationship between $x$ and $y$ appears to be linear or not linear.

The following is a solution to number 10. And were given a pretty small data set of five data values and were asked to make a scatter plot then to say whether it's deterministic or if there's an element of randomness and then we say if it's linear And uh the scatter plot, you would just plot these points like a normal Cartesian plane and that be it. But I always feel like if you have the technology might as well use it. So it's the I-84 is a a nice tool to use if you have it and if you have it you just go to stat and then edit. And here are the data value. So 01 is my first point. So I put all the X. Values in L. One, all the Y values in L. 20125375 11 and 8 17. Now before I actually graft this this kind of looks like it's a pattern. So I'm just Look at this like the rate of change here. Whenever the X. goes up by two The Y. goes up by four. Okay so the same thing here from 3 to 5 goes up by two, the wife goes up by four whenever the X. Goes up by one. So from like 2 to 3 the white goes up by two which is exactly half. Whenever the X goes up by three then the wife goes up by six. So it looks like it's kind of like a 1 to 2 ratio X. X. Increase to Y. Increase. But you know either way if you go to second and then Y equals just make sure this first stat plot is on. So you just click enter and then you make it on and then this first one needs to be selected. So there's the scatter plot. Excellent. Is L. One and wireless is L to at least for mine. Now if you have your data in different columns and you'll need to change those. And then the mark. I like this first mark here because it shows up pretty nicely. So if you go to graph, you should see these data values here and it does look like it from forms. Pretty much a perfect line. Exactly. Perfect line. Now if you don't see those dots, that's fine. It means your windows a little bit different if you go to zoom and then this third option here, I'm sorry, third the ninth option zoom stat. Then you should see that pretty nicely. So it's definitely a linear and this is deterministic. Now I'm gonna kind of cheat here. Um and go to a stat talc and I think it's a linear regression. So X. List is L. One wire list is L. Two. And then whenever I calculate that are square at any time the r squared is one. That means it's it's deterministic. So every value of X. You can predict with a function what the value of why will be perfectly. So that's the answer. There. It is deterministic deterministic because it fits perfectly on a line. Or more specifically every X. There's a pattern. So every X value can predict the Y value 100% of the time. And then is it linear or nonlinear? It's definitely linear. If we go back to the calculator here again. Kind of cheating. But the this is the line. Alright, Y equals mx plus B. Your wife was X plus B. The A. Is to the slopes to that's that rate of change. And then the Y intercept is one. So if you go to Y equals and say two X plus one, it should cut those points exactly. So it hits through all those points. So that's good

The following is a solution to # nine. And were given uh these data values here, it looks like there are nine data values. And we're supposed to make a scatter plot that we did um say whether it's deterministic or there has an element of randomness and to see whether or not it is linear. So what you do is you normally just plot these points on a basic Cartesian plane and then you kind of just look at it. But I always think you know if you have technology you might as well use it. So I'm gonna show you how to do this on the t. I. 84. Uh If you go to stat and then edit, here's where I put my data value. So uh in L. One I put all my ex value, so there's 89 10 all the way up to 16. And then L. Two are my Y. Values to 16 9410 And then back up now before we go further um this kind of looks like it has a pattern if you haven't noticed but these are all increasing by one on the X. Side. And then for the wise those are all perfect squares. So I'm starting to think this maybe quadratic and and it looks like it's actually gonna be a perfect match. Um Before we graph go to second Y equals and just make sure that first staff plot is on the type should be this first type period. That's the scatter plot. And then X. List is L. One and wireless is L. Two unless you put it somewhere else. And then as for mark you choose whatever you want. I like this first one because it shows pretty nicely. So whenever you graph you get to this now if you don't see this these points that's fine. If you just go to zoom And then this 9th options um stab that will show you very clearly what it is and this looks like a perfect parabola. So that's what the scatter plot looks like. And then as for deterministic or random, this is gonna be deterministic. So deterministic um do to it lying and by it, I mean the points perfectly along a quadratic, I spoke quadratic way wrong, squad drat tick function. And uh the reason why my suspicions are here is because these are perfect squares. So another way to phrase that is every X value. You can predict exactly what the Y value is going to be And I'm gonna kind of cheat here a little bit. But if you go to stat and then cowpoke, like I said, I think it's going to be the quadratic regression. So if you go to five excesses, L one wireless is L two. And whenever you calculate this, R squared is one. Whenever the R squared is one, that means we know it's deterministic and here's your actual function. That's X squared minus 24 X plus 144. Or in other words X minus 12 quantity squared. So it is a perfect square. Try no mule actually. So it is deterministic. For sure. And the last part of this is is it linear? Um and very simple. No, it's not. Um It's quadratic for sure. Okay, because it has a nice quadratic function X squared minus 24 X Plus 1 44.

So for this question, we need to remember that slope is the measure of the rise over the run or the change in the Y direction divided by the change in the X direction. So for a the line that we would have would look something roughly like this. So okay, our slope is going to be positive and it's going to be somewhat steep, So out of the possible options that we have, um, the positive options are only Sarah 0.5 or is there a 0.85? So we know that it's a fairly steep lines or slope is going to be 0.85 for be the line that we would have. It's going to look something like this. We see that it's still a positive line, so it will have a positive slope, but it's a little bit less steep this time, so that leaves us with our other option for a slope of 0.5 for C. Our line is going to look something like this and we see that there is no direction to this line, so the slope is therefore going to be zero, since it's not positive or a negative. It's not a positive or a negative slope. Therefore, the slope zero for a D R line would look something like this, and we realize that we're we've got a negative slope that is somewhat steep, which means that our option that we choose is going to be negative 0.45 since that wine was going to be steeper than the one we would get for E, which would look something like this. So that leaves us with 0.30 as our option for E.

The following is a solution number seven. And this gives you a data set. There are 12345678 data values. And were asked to make a scatter plot, er scattered diagram. And then we're going to say whether it's deterministic or random and then we're going to say if it appears to be linear And you can just plot these points like you would on a Cartesian plane. But I always feel like if you have technology you might as well use it. So I'm gonna use the T. I-84 graphing calculator. And if you go to stat and then edit, you can see I put the X values in L. One and then the Y values in L. Two. And then if you go to second Y equals and just make sure that first step plot is on and uh it's just going to it and say it's it's on. And then for the type, choose this first one is the scatter plot. The X list is L. One the wireless dezell too, unless you put it in a different column. And then as for Mark you can choose any of these four. I think the first one's easiest to see though. And then whenever you press graph you should be able to see those data values. Now if you don't see those data values that's fine. You can just zoom in your windows a little bit off which is okay if you just go to zoom and then this ninth option here zoom stat that should give you a pretty clear picture. So this is what your scatter plot should look like. This is you know a computer output but you can do that manually and just make your yourself a Cartesian plane. But these are all the 10. Um Or actually I think there are eight data values there. So that's part A and part B. Is it deterministic or random? So uh this one, it's gonna be random because it doesn't follow an exact straight line and there is no exact pattern. In fact, if we found The correlation coefficient, it wouldn't be exactly one, it would actually be .8 or nine or something. But um that way we know it's gonna be random. So let's go and write that down. There's a randomness element to it element. Yeah, because the points do not form a perfect line. Okay, the x value does not 100% predict the y value, so there is a randomness to it, and then it asks us is this linear? And it does seem linear. There's a linear relationship. So it seems there is a linear relationship, and you can just determine that just by going back to that scatter plot, it kind of looks like this is a, you know, a nice straight line, there's very little deviation, it's not, you know, a circle or it's not going up and down, it's pretty much just going up from left to right, so this one appears to be linear.


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