let-overlinexfrac1nleftx_1cdotsx_nright-and-overlineyfrac1nlefty_1cdotsy_nright-show-that-the-least-

Question

Let $\overline{x}=\frac{1}{n}\left(x_{1}+\cdots+x_{n}\right)$ and $\overline{y}=\frac{1}{n}\left(y_{1}+\cdots+y_{n}\right) .$ Show that the least-squares line for the data $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ must pass through $(\overline{x}, \overline{y}) .$ That is, show that $\overline{x}$ and $\overline{y}$ satisfy the linear equation $\overline{y}=\hat{\beta}_{0}+\hat{\beta}_{1} \overline{x}$ .[Hint: Derive this equation from the vector equation $\mathbf{y}=X \hat{\boldsymbol{\beta}}+\boldsymbol{\epsilon} .$ Denote the first column of $X$ by $\mathbf{1}$ . Use the fact that the residual vector $\epsilon$ is orthogonal to the column space of $X$ and hence is orthogonal to $\mathbf{1} . ]$

Angelo Rendina · Accepted Answer

we can write a genetic regression line problem in this way where we have on the first column the wise off the points. Then we have the big Matics X, which is the mother lode of coefficients, where we have in the first column all once and in the second columns the axis of the points that is my exact multiplies the Matics Beetle, which contains the parameters that we want to find a better one and bitter, too. And then we have the air or the residuals now to solve the problem. Let me just write a the metrics one over end. Well, all ones in a row. And we upset. Right now that is one over end. The first column of acts. Well, transport&#x27;s because we put it as a raw and we&#x27;re going to use this observation later. So now let&#x27;s multiply from the left by a because remember that why bar and X bar are the average is off the points and well, when we multiply from the left by A were basically obtaining the means so well different from the left. The previous equation we have a wise equal to X beat up, plus a Absalon. An hour&#x27;s observed on the left hand side. We know that it&#x27;s just why bar So the average of the wise and on the right inside will a X is going to be the vector. Well, the first entries one because the average is awful ones. And then the second column is the average of the axis. So just x bar. And then that multiplies the metrics beetle, which is better one bitter too, and then plus a Absalom. And so basically, the point is, we want to show that a absolute zero. And if we do that, the proof is over. So we know that the residuals saw Absalon is or for gonna to the columns off X or other way of writing that is that a genetic column of X. We put it, transports and we multiply by Absalon. We should get zero. But we observed that the first column Oh X is a so a upside down. He&#x27;s one over end. The first column of Ex Put as a row so transposed the multiplies Absalon and therefore that zero. Therefore from the previous page, we see that why bar is equal to be one plus beetle to x bar, which is we wanted to show

Angelo Rendina · Answer

That&#x27;s right, The medics of the coefficients Big Axe as the first column is all ones and the second column has the axes of the points X one Exxon. And we know that some of X, which is what I did Note X one plus plus Exxon and zero. So the point starting mean deviation form We want to show that Well, X transpose X is a dagger on metrics. So let me call the first column of X C one and the second column of Ecstasy too. Now X transpose x Well, that&#x27;s see one transport see to transpose. But as Rose and then C wants you to thought, these are two by two matrix and well, we have C one transports you want you want transparency to and then see to transport C one and C to transport, see to All right, no c one Transparency One is just well, some of ones. And we have exactly in data points. So it&#x27;s n them for C one trust. Beyonc? We see that that the some off ex multiply by one so there&#x27;s just some of X and similarly c two transposed. The one is some of ex again and then see to transport see to it the sum of the squares. So some X squared. But now we&#x27;re done because we know that some X zero. So we have an 00 sum of X squared, which means that X transpose x. He&#x27;s a diamond.

Foster Wisusik · Answer

Okay, This question wants us to show that the condition m n equals negative one, which is our normal condition for perpendicular lines. We want to show that that implies that the direction vectors of the two lines is equal to zero. So we&#x27;ll start with the fact that dividing this, buy em, we get that and is equal to negative one over M. And that&#x27;s what we learned back in algebra, where perpendicular lines have opposite reciprocal slow. And now we&#x27;ll use the fact that our two lines are going to be MX and and X, and the constant does not really matter here because the direction vector of the line is going to be the same for each, it just might be shifted. So the direction for the first line which will call you well, this vector for the wine is in the direction of one m because you move em units up for everyone, step in the X direction and then similarly V is one comma end. And then if we take the dog product of these two vectors, we get you dot V is equal toe one plus m and or the orthogonal ity condition. We started with Eman equals negative one. So if this condition happens, then these two factors air perpendicular, which is exactly what we expect.

Ahmed Kamel · Answer

at Problem number nine, we are given to system equations. The first is NB equals sig mix onto applied by em that is equal boosting July. It&#x27;s called this question number one, and the question number two is sig makes not like by B plus. Seek banks squared, multiplied by em equals Seek Matics. Why it&#x27;s called this wish number. So at first, let&#x27;s work with the first equation. We will divide you. Fight the first equation by and so we will have me plus seek makes to like buy em over End equals sigma Lie over in So it&#x27;s separate. Be at one side. Swoopy equals Sikma y over any minus seek makes I want to buy by AM over any which is equal to it&#x27;s heavy here in and here we will have Sigma. Why minus him speak makes, which is commonly used equation for Be Now for a question Number two. It&#x27;s substitute the value off me by this way. So in equation number two, we have sick Max B plus Sigma X squared I m equals sync makes what So we will have sig makes multiplied by sigma y minus. I m sick makes over any and where you will have Sigma X squared multiplied by em plus seek makes square multiplied by em Equals equals sig makes what? Okay, so now we can separate this into two values. We haven&#x27;t Sigma X. I think my wife over in minus I m sick. Max, hold word over in Plus, Sig makes Squirt want to fight by em equals my ex wife. So let&#x27;s have all the terms that have em in it in one side. So at first I would say we eat here we have Sigma x Y minus sig makes by sigma lie who were in. So we moved this value to the other side and here we will have a common factor off em. And we will have Sigma X squared, which is this value minus. We have them here. So you have Sigma X old squirrel over in. Okay, so we can say that and is equal to this. Value is we will have to unite the denominator at n so here we will have and Sig Max. Why minus seek makes by Sikh Malai And we will do the same for this value at the denominator. So we will unite the denominator over the denominator at M. So we will have here in Sigma X squared minus sick makes old squared and can cancel this end with this end. So the M value would be equal to And I think my ex wife minus seek makes by sigma Why over n sigma X squared minus big Matics old squirt which is commonly used value for and

Cullen Miller · Answer

hope you&#x27;re doing well. So for this problem, we have the points. 13 25 36 56 and 79 So here are two equations that we have to get our eight, and the values for our linear are linear equation that fits this data best. So first, we&#x27;re going to start with, um, getting some of our son values like some of XK, some of like a and things like that, so it can just plug it straight. So some of XK just going to be the sum of all our X values in these points, that&#x27;s going to be one plus two plus three plus five plus seven plus choose 33 Pipe 7 12 The mysterious six that gives us 18 It&#x27;s a summer X K is equal to 18 and then I&#x27;m going to figure out some of why K as we can see right there. So some of why Kate is going to be some of all our Y values, which will be three plus five plus six plus six plus nine three plus five is a six plus six is 12 plus nine plus faultless nine is 29 said 29 is the sum of all of our Y values. And then now we want to find the sum. This right here, the sum of all of our X value squares. The sum of X k squared is just some of each X value squares. It&#x27;s going to be one squared plus two squared plus three squared, plus five square US sudden squared simplifying that that is equal to when Skirt is one to squirt us for three. Squared is nine by scored just 25 7 square. Just 49 one plus four plus five gives us 14 waas 25 plus 49. Is this 74? This gives us 88 so 88 z equal to the sum of our all of our X Values Square. And then the last thing that we need to find is this the symbol of our X and Y values and multiplied, but by each other&#x27;s We&#x27;re going to take three times. One was five times two and etcetera add them all together so some of all of our XK like a will be equal to we&#x27;ll be one times three plus two times five plus three times six Waas five times six plus seven times nine So simple. Find this through times one is three plus five times two is 10 plus three times six is 18 plus five times six is 30 plus seven times nine is 63. We plus ton is 13 plus 18 plus 30. This 48 plus 63 13 plus 48 gives us 61 63. That&#x27;s equal to 1 24 That means that well, what&#x27;s 1 24 is equal to the sum of all the XK. Why can&#x27;t so we have each of our values for Sung&#x27;s right here so we can just plug these values into this equation. First, we have some of all exes, which is 18. What&#x27;s gonna give us 18? A. 18 A plus? We had N B, and it&#x27;s just the number of points that we have. Such five since five b is equal to see. It&#x27;s the sum of All Wise, which we got 29 for that 18 8 plus five years. 1/4 29 moving after a second equation, we have the sum of all X K squared, which we got found out was 88 said e v a waas. We know we have some of all XK, which is you got before. That&#x27;s just 18 since plus it&#x27;s you be that&#x27;s equal to on the sum of all exes and wise multiplied by each other, which we got was 1 24 All right, so this right here is our system of equations that we want to try and solve. So, solving this system of equations, we&#x27;re going to try and use the elimination method. So we were going to use the elimination methods. We&#x27;re going to try and eliminate our be variable. So to do that, we&#x27;re going to multiply this first equation by minus 18. Let&#x27;s by the second equation by five news It&#x27;s why you by five. So distributing this, uh, go into our first equation distributing this minus 18 have minus 18 tons. 18 a gives us minus 3 24 a minus 18 cents minus five b gives us minus 90. That&#x27;s equal to minus 18 times 29. Gives us minus 5 22 and then for a second equation we have five times 88 a gives us or 40 A. We have five times 18 b gives us plus 90 B That&#x27;s equal to five times 1 24 which gives us 620. It&#x27;s adding these two equations together right here we have minus 90 B plus 90. Be these. Just cancel out and go to zero. So this we&#x27;ve eliminated are variable B. So you&#x27;ve got minus 3 24 8 plus 4 48 That gives us 1 16 A. That&#x27;s equal to minus 5 22 plus 6 20 Just 98 been dividing both sides of our equation. By 1 16 we end up with a is about equal to 0.845 So now we want to try and figure out what you value this corresponds to, and I But it would be valued this correspondences. We can put this all in back to our first equations. We can rewrite our first equation. Here, 18 8 plus five is equal to 29. They can solve this equation for be plugging are a value to solving this equation for being. We have five b is equal to 29 minus 18 A. Then we know so we can divide both sides of the equation by five we end up with is equal to 29 minus 18. A over five A is approximately equal to 0.8456 equal to 29 minus 18 times 0.84 5/5. And when we simplify this, we end up with B being approximately equal to 2.759 So we know our formula for this line. Win your line. That&#x27;s that&#x27;s the least squares line is why is equal to X plus B got our A B values. Here&#x27;s that means our line is why is about equal to, besides that? Why it&#x27;s approximately equal to 0.845 x plus 2.759 This is our final answer for a least squares line for this data so we can plot this. And if you put this on a calculator, you can see that it&#x27;s the line. Fits the data pretty well Ends also, if you use the calculated, determined police scores lineman, it should get the value. Something like this. Let&#x27;s plot this to see how accurate the state is. So I&#x27;m just gonna ought the first quadrant because that&#x27;s where all our values are located. So we have, um, do a little bit better graph. So you see where values come in. So be up to nine. Yeah, these by ones. All right, Sara, data are we have the points 13 about their You have the 0.25 horse once about right there with 36 which is right there with 56 We have the points seven nine right about there. And then we know if we graph this data. So we&#x27;ve graft our data and then are aligned. Why is approximately equal? 2.54 x plus 2.759 Serve. Why intercept is going to be about their The graph goes, you&#x27;re rough. Estimate the graph of the graph ghost graph. It looks something like that so you can see that that&#x27;s a pretty good fit for this day because it goes right in the middle of all these data points, and it&#x27;s kind of the average of all these data. So this is this is our shows that our line of best fit that we obtained is a pretty good line of best. Are pretty good at least squares. Um, so at least squares, like so a regression line. So this is our final answer. Then we should be about good. We have our, um, equation for a least squares line, and we&#x27;ve also sketched her points to show that the line does is a pretty good representation of the points of this fit the points pretty well. So and again. If you use your calculator online calculator that term in what the line at least squares line will be It should be something very similar. So should be about good. So thanks. And I hope that helps.

Fuzail Shakir · Answer

for this question. We know that we have The sum from J is equal to one up to em off X comma j of Except Jae Weiss of J. And this is equal to m times. The sum from M times sum from J is equal to one up to end off except J squared X subjects were minus B times the sum minus B times of some for O. J from one end off excess of J. Now, similarly, we&#x27;re also gonna know we also have to find a wife except I So we have the sum from eyes equal Thio one up to end off Wise of I is equal to m times of some from one up to end sometimes some from one up to end over x sub I plus x of I plus n times beaten

Problem

Derive the normal equations $(7)$ from the matrix…

Problem 14 Medium Difficulty

Answer

$\sum y-n \hat{\beta}_{0}-\hat{\beta}_{1} \sum x=n \overline{y}-n \hat{\beta}_{0}-n \hat{\beta}_{1} \overline{x}$

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$\sum y-n \hat{\beta}_{0}-\hat{\beta}_{1} \sum x=n \overline{y}-n \hat{\beta}_{0}-n \hat{\beta}_{1} \overline{x}$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Caleb E.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Anna Marie V.

Caleb E.

Samuel H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications