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In Exercises $21-24, \mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ are noncollinear points in $\mathbb{R}^{2}$ and $\mathbf{p}$ is any other point in $\mathbb{R}^{2} .$ Let $\Delta \mathbf{a} \mathbf{b} \mathbf{c}$ denote the closed triangular
region determined by $\mathbf{a}, \mathbf{b},$ and $\mathbf{c},$ and let $\Delta \mathbf{p} \mathbf{b} \mathbf{c}$ be the region determined by $\mathbf{p}, \mathbf{b},$ and $\mathbf{c} .$ For convenience, assume that $\mathbf{a}, \mathbf{b},$ and c are arranged so that det $[\tilde{\mathbf{a}} \quad \tilde{\mathbf{b}} \quad \tilde{\mathbf{c}}]$ is positive, where $\tilde{\mathbf{a}}, \tilde{\mathbf{b}},$ and $\tilde{\mathbf{c}}$ are the standard homogeneous forms for the points.

Let $\mathbf{p}$ be any point in the interior of $\Delta \mathbf{a} \mathbf{b} \mathbf{c},$ with barycentric coordinates $(r, s, t),$ so that
$$
\left[\begin{array}{lll}{\tilde{\mathbf{a}}} & {\tilde{\mathbf{b}}} & {\tilde{\mathbf{c}}}\end{array}\right]\left[\begin{array}{c}{r} \\ {s} \\ {t}\end{array}\right]=\tilde{\mathbf{p}}
$$
Use Exercise 21 and a fact about determinants (Chapter 3 ) to
show that
$r=(\text { area of } \Delta \mathbf{p} \mathbf{b} \mathbf{c}) /(\text { area of } \Delta \mathbf{a} \mathbf{b} \mathbf{c})$
$s=(\text { area of } \Delta \mathbf{a} \mathbf{p} \mathbf{c}) /(\text { area of } \Delta \mathbf{a} \mathbf{b} \mathbf{c})$
$t=(\text { area of } \Delta \mathbf{a} \mathbf{b} \mathbf{p}) /(\text { area of } \Delta \mathbf{a} \mathbf{b} \mathbf{c})$

Yeah. All right guys, the first thing we need to do here is we do identify on are null and alternative hypothesis are no hypothesis. Excuse me being that the variations are going to be the same. And the turn of prosthesis is that they're going to have different variations as you can see right here. Okay, so the next thing I did to solve the problem was I opened a google she and I am putting a lot of the data. You'll notice all the data is and put it right there. Okay. And then, so what I'm gonna do from there, what I'm gonna do from there is this I'm gonna go ahead and calculate the variances for each. The first variance I'm gonna calculate using google sheets. So I'm gonna type equals var open parentheses and then just highlight everything I want. I know have an extra box in there. That's not a big deal. And there is my first variant. Second variants will be found the exact same way it looks like it already knows what I want, which is great. And there is our second variance. Alright, so from there I need to find my f statistic. Okay, so my F statistic is just going to be the greater variance divided by the smaller variance and there we go, there's my f statistic. Once I found my F statistic, I'm ready to find my P value. Actually. Sorry, scratch that. Once I find my F statistic, I'm almost ready to find my P value. But there's one more step in order to find a P value. Using google sheets with an F statistic. First need to calculate degrees of freedom. Okay, very easy to do in google sheets. So degrees of freedom is just all the data, the amount of data you have minus one. So how do you find that? Google sheets is equals count. Just highlight everything. I have noticed how I do have an extra box there. That's not a big deal to subtract one. And that I have 20 in this case 24 degrees of freedom. And in our second case again equal count Open parentheses, highlight everything and I do need to subtract one. And the other case have 15 15 degrees of freedom. Okay, so now I'm ready to find my P. Value. So my P value uh through here is going to be something called an F. Distribution or F. Dist. And then open parentheses. I need to put three values. Number one, my F statistic. The second value is going to be the number of entries from S one squared. I'm sorry the degrees of freedom from s one square in this case is 24. And my last and final value is the degrees of freedom from my S two squared. So that's right here. I close all that up and there is my P value. However, I'm not quite finished yet. Okay, that is a P value for a one tailed distribution. What do I mean by that? That's a good P value. If that was a greater than or less than sign, this is a not equal to sign. Okay. That's not equal to sign. Which means um uh that which means that I have actually had the wrong P value right now. Since this is for a one tailed distribution and I need a two tailed distribution because they're not equal sign, I need to take this value multiplied by two. So equals I click on the value multiply symbol in google sheets. Asterisk hit A two and I'm done. That is my P value. Notice that P value is just a touch above 0.05, meaning I'm going to fail to reject the no hypothesis. I'm gonna fail to reject. And all hypothesis

Okay for these problems were given a formula to follow. It's in be plus, Sigmund eggs, which are some of the X values times, eh equals the sum of the Y values. The second equation we have is that some of the X values again times be again plus the sum of the X value square times a A and that's gonna equal the son of the product of the X and my values. So maybe this is saving you some time, but I took the time to add all these things up s o in the number of things we have, the number of points is eight. If we Anna Bolivar X values, it's gonna give us 36. If we add a bolivar, why'd I use we're gonna get 30.3. If we add up all of the squares of our X values, it's gonna give us 204 And if we had it all about the product of our X and y eyes, that's gonna give us 179.4. Now, we just need to solve this system of equations, so I'll do that in the next slide. Okay, so we're gonna have eight b plus 36 A is equal to 30.3. On our second equation, we're gonna have 36 being some of the exes again. That's two of four A connects equal to 1 79.4 So we use a process of elimination on these, and it's about the values, Um, where I can make the them both into 72 to have equal coefficients. So we'll try my top equation by nine in my bottom equation by two. Okay, so that's gonna be 70. Be a start. Pullin equation nine times 36 is 3 24 nine times 30.3 is to 72.7. Care second equation. There's our 70 to be again. This is gonna be 48 Hey, and then 3 58.8 Notice My second equation is larger than the first equation for the other coefficients. So I'm gonna say, Trent, my top equation from the bottom equation, that might seem odd, but really, it's just a change. A prospectus are the values or canceling out. Faraway minus 3 24 is 84 a 3 58.8 minutes to 72.7 is 86.1. If I divide both sides by 84 on a his 1.25 A lot nicer than I thought it would turn out. So it is 1.25 Plug that back into my first equation. So have eight B plus 36 times 1.5, equaling 30.3. 36 times 1.25 is 36.9. If I should try that from both sides, I have a baby is equal to negative 6.6. I met him on both sides by eight. Be value is gonna be negative. 0.8 to 5. Okay, so the least squares regression line. This is gonna be why equals 1.25 acts minus 0.8 to 5. You can use a calculator to confirm this. Um, disappointment blank your values into list one list, too, and then do a linear regression analysis. Um, this is, uh it works. I double checked it. Um, so this would be the value of your line. You could also on your cock. It comes to the line of best fit. You can go to plots above your y equals on T I instrument and turn plots on with list one list to selected. And that should give you a a pretty nice view of what this line of best fit would look like. Thank you.

Okay. All right, guys. So in this one, um our uh our research objective, I'm sorry, our research objective is this question is music cognition related to cognition is pertaining to abstract operations, such as mathematical or spatial reasoning. That's our objective is to answer that question. Our sample is the 36 college students who participated in the study. Um are descriptive statistics. Are the average score after listening to Mozart was 1 19, while the average score, falling silence is only 1 10. And so we will conclude that subjects performed better on abstract and spatial reasoning tests. I'm sorry. Test Thank you after listening to Mozart.

For the match pairs given on the right. We want to conduct a sign tested matched pairs testing P does not equal 0.5 at alpha equals 0.5 significance. This question is testing our understanding of non parametric tests in particular how to conduct a scientist of matched pairs. We proceed there steps A through D below to solve. So first, in a we stayed alpha hypotheses, this gives alpha equals 0.5 H. And r P does not equal five. H. A P does not equal five. So the no hypothesis P is equal to have, the alternative is not A and B. We compute the test out. So are signs for these matched pairs are as follows and equals 12 is the total number of matched pairs. So X equals the number of plus or the total number plus and minus which is 4/12. Thus we have seen it equals x minus 5/45 over and equals negative 1.155 Thus the p value is for a normal distribution, P equals two PZ greater than zero equals 248 Thus we conclude that we fail to reject asian off because he is greater than alpha, which means that we lack evidence to support the claim P does not equal five.


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