find-the-least-squares-line-ybeta_0beta_1-x-that-best-fits-the-data-20-100214-and-24-assuming-that-t

Name: find the least squares line ybeta_0beta_1 x that best fits the data 20 100214 and 24 assuming that t
Uploaded: 2019-08-15
Duration: 2 min 5 s
Channel: Numerade
Description: Find the least-squares line $y=\beta_{0}+\beta_{1} x$ that best fits the data $(-2,0),(-1,0),(0,2),(1,4),$ and $(2,4),$ assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.

Question

Find the least-squares line $y=\beta_{0}+\beta_{1} x$ that best fits the data $(-2,0),(-1,0),(0,2),(1,4),$ and $(2,4),$ assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.

Angelo Rendina · Accepted 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.

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

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

Alayna Abraham · Answer

in this problem, we&#x27;re going to try and find the least square line or another way to put that is the best fit line of our linear model. And so that means that this equations will be, ah, in the form of y equals MX plus B until to solve it. We should use thes two formulas that I&#x27;ve given here, and I&#x27;ll just break down some of the parts in it So n equals the number of points that have been given. So and this problem, we have been given four points star on his equal to floor, and this symbol just means summation or some of on. So let&#x27;s get into it. So let&#x27;s are for Let&#x27;s sell for em. First up is equal to end, which is four times the sum of X times y. So we have to multiply all our X and Y values and add them together. So two times five is 10 0 times negative, 05 times 3 15 and one times negative. There is next three Sobel sum everything up and we&#x27;ll get 22 for this cart, and the next we subtract it from all the some of our X values. So two plus zero plus five swan is equal to eight times the sum of our Y values. So five plus night of one plus three plus negative three is equal to four, and that is all over four times the sum nation of X squared. And so this means we have to take all the X values and square them before we add them. So we have to do for certain, which is four plus year square for just zero flared, which is 25 plus one squared, which is one. And so we get 30 minus the summation of excess school. The whole thing squared, so this part is a little different. We take the sum of our X values first, which we&#x27;ve already done, and we&#x27;ve gotten eight for that, and we square that will thing which is 64. And when you sold it all out, you should get four times 22 which is 88 minus 32 all over 120 minus 64 on that is equal to 56/56 which is one which is nice. That&#x27;s a nice number to get. And then let&#x27;s help for be so b is equal to us some of our y values. So we&#x27;ve already solved that above. And that&#x27;s four minus. Excuse me, sir, R M value, which is one times the sum of our X values, which is eight again. We already saw that above Haulover four. So when you solve it out for minus eight over before you get negative one and if we then put it into our linear model will get why equals X minus one and that&#x27;s dancer.

Problem

Suppose 5 out of 25 data points in a weighted lea…

Problem 1 Easy Difficulty

Find the least-squares line $y=\beta_{0}+\beta_{1} x$ that best fits the data $(-2,0),(-1,0),(0,2),(1,4),$ and $(2,4),$ assuming that the first and last data points are less reliable. Weight them half as much as the three interior points.

Answer

$2+\left(\frac{3}{2}\right) x$

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Calculus 3 Educators

Lily A.

Heather Z.

Samuel H.

Joseph L.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 6

Video Transcript

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

Linear Algebra and Its Applications

Lily A.

Heather Z.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$2+\left(\frac{3}{2}\right) x$

Linear Algebra and Its Applications

Lily A.

Heather Z.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Heather Z.

Samuel H.

Joseph L.

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

Linear Algebra and Its Applications