find-the-equation-ybeta_0beta_1-x-of-the-least-squares-line-that-best-fits-the-given-data-points-011

Question

Find the equation $y=\beta_{0}+\beta_{1} x$ of the least-squares line that best fits the given data points.
$(0,1),(1,1),(2,2),(3,2)$

Pawan Yadav · Accepted Answer

Okay, So question is we need to find best fit line using the square method. And the model is why is equals, Toby Bitter not plus bitter one time sex and the points are given to us. So let&#x27;s assume a solution is metrics. Form is ex Bitta equals two. Why? Yeah, your exes metrics representing coefficient off beat or not, and coefficient of b town. So since bitter not coefficient is one so it will be one. And here proficiently sex. So that is zero right. So 10 next will be 11 12 and 13 So this is our ex. And why is the output that is represented by this value? So that is 112 to Okay, so now we have all the datas. And here Vita is nothing but beat or not And Vita one. Okay, so now we can substitute that in our metrics equation. So we have 10111213 times beater. So we&#x27;ll write it as beat only. Here we have 1122 Okay, So now to solve this, we need to have a square metric so we&#x27;ll multiply by x transpose a pre multiplying. So x transpose will be 10111213 right times 11110123 Vita is equals to 10111213 times 1122 Okay, so now we can solve this. Okay, so this row will be multiple, er with this column so we&#x27;ll have one times one plus one times one plus one times one plus one times one writes thing. Okay, so that gives for second on This will be multiplied with this and adding it will give one plus two plus three. So that this six then multiplying this with this one. So that will give us again. Six and 0123 multiplied With this one will give us zero plus one square plus two squared plus three squares on nine plus 4, 13, 13 plus one. For them times bitta is equals toe. Now we can multiply this and this value. So one times one all this one little gives him and I think it really one plus 1 to 2 plus 244 plus 26 And this into this gives us zero plus one plus four. That is five plus six. So it is 11. Okay, so now we have our metric situation and it is square metrics. So now we can sort beetle will be 466 14 in wars times 6 11. Right now in worse. Off these metrics will be 14 times four minus six times six. And we know we just have to interchange this 14 and four diagonally and we have to interchange this with minus sign. Right, So this will be minus six and minus six. And this is six and 11 on multiplying. This time we get 20 and multiplying this we get 18 and this is it. Okay. So arbiter not and beat a one is equals toe 18 by 20 which is 200.9 and eight by 20 inches. 200.4 therefore a model list. Why? It calls 2.9 plus 0.4 x on that is answer. Thank you

Angelo Rendina · Answer

we know that the problem of finding the regression line is essentially a problem off the square solution. So we&#x27;re given the data, which are the four points. So we formed the Medics Axe, which has four rows as many as the data we have. And on the first column, we have all ones because these corresponds to the constant term better. Zero. And the second column we put the ax is off the points they were given. So 124 and five and then for the constant term we have the wise of the points. So 0123 Let&#x27;s call these metrics. Why? So since we need to solve the least square solution, we know that we need to form the normal equation for it. So let&#x27;s start computing X transpose x a simple computation. It&#x27;s a two by two matrix with entries 4 12 12 46 And then we also need to compute X transpose. Why? Which is a two by two matrix? Well, to buy ones are column vector with entries six and 25. Now that we have these data, we can just compute the least court solution because you know that are solution Beata It&#x27;s going to be even by x transpose x inverse that multiplies X transpose y and we have completed these medicines in the previous page. So x transpose x inverse is one over. It&#x27;s the terminal which turned out to be 40 then 46 4 months storm on a cell phone on the second and I are gonna and then the other term in green is just X transpose y, which was 6 25 Now it is just initial matrix multiplication. So one over 40 and the resolve to year is minus 24 28. After the vote, anybody 40 we finally got at zero point was minus 0.60 point seven and this is the least squares beata And so these are the conditions of the regression line. So we know the constant term bit zero. He&#x27;s minus 0.6 and the slope beta one is 0.7

Angelo Rendina · Answer

finding that aggression line is essentially a least squares problem, and we know that to solve that, we need to form the metrics acts of the coefficients, what we have as many rose as the points we have. So we have four O&#x27;s and the first column is all once and on the second column. We have the access off the given points, so we have 235 and six. And then for the constant term little why we have again for Rose as many as the points. And we put the wise of the given points or 3210 So says we need to compute the normal equation. We started by computing ex transfer, relax and x transpose y x transpose x Just a simple computation. Pistons out to be 4 16 16 74 and then X transpose y is 6 17 Now that we have everything, we can compute the equation. So we know that bottom is given by X transpose x inverse that multiplies extracts was why that we have just computed so in red we have X transpose x inverse, So one over determinant. They&#x27;re multiplies, said the four months, 16 months 16 4 and then in green, we have X transpose y so 6 17 He&#x27;s just an issue of multiplication. So one over 40 that multiplies the metrics 1 72 minus 28. So all in all, beta turns out to be 4.3, minus 0.7. That means that the confusions that we&#x27;re looking for are bitter. Zero is 4.3 and bitter one is minus 0.7.

Alayna Abraham · Answer

Okay, So in this problem, you&#x27;re trying to find at least squares line Another way to put this is the best fit. One linear model. And because it is a linear model, it&#x27;s gonna take The answer will be in the form of why equals Sorry y equals MX plus B. And the easiest way to solve that is by using the formulas ever in here. And I&#x27;ll just quickly go through what some of the variables mean so and just means the number of points and in this way, have four. And if you do it with the symbol means it just means some of I&#x27;ll probably be referring to it as some summation, which just means the same thing. And the rest I will explain as we solved. So let&#x27;s get into the problems so m equals and four times the sum of our X times y values. So zero times three issue plus one times negative one is negative, plus two times six is 12 plus floored time sixties 24 on. Once you take the sum of everything, you&#x27;ll get 35. It&#x27;s a very but doesn&#x27;t hear minus some of our X values. So cereal plus more close to close for which is equal to seven. So seven times the summer y values, which is three, wasn&#x27;t one plus six plus six and that illegal to 14 of teams pull over and again. So four times the sum of our X some of X squared. So this time we have to take all our X values and square it before we can add them up. So is yours. Go to zero plus one squared is one plus two squared. This floor says four squared is 16 and once you have them all of his 21 minus the summation of X holding square so that we bought it in the summer of X above 37 So the whole thing sport is 49. Make sure you pay attention to where the squared is being placed, whether it&#x27;s inside or outside the parentheses, because that will change the value off what you&#x27;re solving for. And you should be careful that that can be a reason why you don&#x27;t get this answer. Correct. And when you do the math, you&#x27;ll get 42/35 simple by six by and if you want to industrial form. It is 1.2. I will be using decimal form. Um, but if you have these fractions, use fractions with yourself in for be so B is equal to the sum of our Y values, which we again were the salt in the in formula. So 14 minus and so one point you and some of X, just seven over full. So we get points for And then that means our answer is why 1.2 x plus 1.4 scenes.

Alayna Abraham · Answer

and this problem, we will be solving for the least squares line or another way to put that is the best fit line of linear model on because it is a linear model. It should follow the question. Why was I&#x27;m explicit? Piece or Inter should be in the form and you can easily solve that, using the formulas that I put in here and so I&#x27;ll just go through some of the variables, um ah, symbols that are in this formula that you&#x27;ve been given so and it&#x27;s just equal to the number of points. And in this problem, we&#x27;ve been given three. And if you ever see this symbol, it just means Summer on and the rest I feel, are pretty self explanatory, or I&#x27;ll explain it as I go through the problem. So let&#x27;s start with First on and is people to end, which is three times the summation of X times. Why so first we before we some up the values we need to multiply our x times y so two times negative t one tons Native three is three e. I just assumed everyone knows that you&#x27;re a time 00 So negative seven goes in here minus the summation of our excise two plus 103 And then our some of our wife fires is minus two plus Nate Dogg ASIO, which five all over three again because again and is are equal to three and then the summation of X squared. Now notice where the squared is in this part of the problem. It is inside the parentheses. So that means we have the square each x value before we Adam. So two squared is four plus one squared. It&#x27;s one and see was greatest. Eurostar value is five and then subtracted from that we have to do the summation of X on the whole thing&#x27;s cleared. So that means we have to find the summation of X, which we already did above and that is three on. We have the square, which is now not now. Notice where the square is taking place. Either insider outside the parentheses and make sure you have taken that into account on. When you solve the problem, you will get negative 6/6. So ATM is equal to negative one. And now that we found and we consult for fee so musical to the some of y, which we body self for love. So that&#x27;s negative. Five minus negative one. So that becomes plus one times X, the sum of X, which again we&#x27;ve also solved above to be three all over our and value, which is three. So that&#x27;s negative. 2/3. So then our answer is why 2/3 so minus 2/3.

Alayna Abraham · Answer

Okay, So in this problem, we&#x27;re trying to find the equation for the least squares line or another way to put this is we&#x27;re trying to use the points that we think given to find a best fit line that fits our linear equation model, which is why equals MX plus B. The simplest way to do that is to plug into these formulas here. And so let&#x27;s break this formula down. Let&#x27;s break the first The M for Ladan S. O. N is just equal to the number of points. And in this problem, we have to. And if you don&#x27;t know what this symbol means, this just means the some mation up, and that just means some of so let&#x27;s start answering the problem. So let&#x27;s sulfur m first, because if you notice you need em in the second part of the relation to pine, be so m equals two. And just to times the summation of X times Y on, I find that if you cut because there are so many different parts to this equation, if you write under each um value under each point what you want, then it makes it easier to solve out do an example of that. So for the first part, because it is the multiplication of X times why we first have to do six times three, which is 18 and negative to Time zero, which is hero. And because we&#x27;re taking the summation, it&#x27;s the sum of so 18 plus zero equals 18. And so that&#x27;s what we have for the first part. And then the second is the sum of the X failures. So six plus negative to People&#x27;s Toe four and then for why it&#x27;s three plus zero is equal to three. So now we have the top part done. And then again, R N is equal to two, and this next part is X squared. So the summation of X squared. So that means we have to do six Squared, which is 36 plus negative two squared, which is four. And so we get 40 when we sum it up. And then, um, the next part. It&#x27;s the summation of X squared. So don&#x27;t mix these two up. They are different this time. We&#x27;re taking what we got for the sum of X and were squaring it. So that&#x27;s the floor that we got here were squaring that. So that&#x27;s 16. And so then once you do the math out, So 36 minus 12 and 80 minus 16 you find that it simplifies down to three ace. So that&#x27;s what our M and then to solve for B. We have to take the summation of our Y values. And we found that earlier right here. So that&#x27;s three minus three ace, which we just saw four times again, this summation of X, which we already saw four to be four and then all over, too. And when you simplify and do your math out, make sure you do the math out correctly. I know it can be tricky. Sometimes you get three fours, and so now we can put our equation together. So now our equation is why equals three X Plus three Force and that the answer to this problem

Problem

Find the equation $y=\beta_{0}+\beta_{1} x$ of th…

Problem 1 Easy Difficulty

Find the equation $y=\beta_{0}+\beta_{1} x$ of the least-squares line that best fits the given data points.
$(0,1),(1,1),(2,2),(3,2)$

Answer

$y=0.9+0.4 x$

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

Linear Algebra and Its Applications

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$y=0.9+0.4 x$

Linear Algebra and Its Applications

Catherine R.

Kristen K.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Kristen K.

Samuel H.

Joseph L.

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

Linear Algebra and Its Applications