a-certain-experiment-produces-the-data-118227-334438539-describe-the-model-that-produces-a-least-squ

Question

A certain experiment produces the data $(1,1.8),(2,2.7),$ $(3,3.4),(4,3.8),(5,3.9) .$ Describe the model that produces a least-squares fit of these points by a function of the form 
$y=\beta_{1} x+\beta_{2} x^{2}$
Such a function might arise, for example, as the revenue from the sale of $x$ units of a product, when the amount offered for sale affects the price to be set for the product.
a. Give the design matrix, the observation vector, and the unknown parameter vector.
b. [M] Find the associated least-squares curve for the data.

Pawan Yadav · Accepted Answer

Okay, so questioning certain experiment produces the data is given. I&#x27;m going to describe a model that produces a least square best feet for a model of equation as y equals two bitter one X plus Bitter two X Square. We need to find design metrics, observation, vector and a non parameter vector. And also the associative equation with that. Okay, as the model given is y equals two beat a one x plus Beta tau X square. Okay, so for every point for why one, let&#x27;s say four x one y one pair, we&#x27;ll have why one is equals to beat a one x one plus Peter too x two squares. Similarly, why two will be better to x two plus Sorry, Peter one x two plus B tattoo X two square Why three will be beat a one x three plus bitta two x three square and it will keep on going. Okay, so this can be modeled in a metrics as right. So we&#x27;ll write this in left hand side because this is our input and this is a output. Okay, so we can write. This is X one x two squared. Sorry. Excellent squared x two x two square x three extra square x four x four square and we can go till exonerate Extent square. And here this will be multiplied with beat a So that is, beat a one and vita too. Okay, so this is the left hand side. Right Inside is very quickly y one y two y three. Why? For right And okay, so this is a model now. So part one is done. We formed the model. We can substitute the values here, the given values that is so this can be returning us. Therefore we have x one x one squares or one one one square. So that is one only then to to square so to four. Let me write directly the square values then 33 square 44 square. Then we have five. Five square. Right? So upset we don&#x27;t have this one right. We have four and 38 Okay, so this is our model. Then beat a one b Tartu we need to find. And again why one y two is given to us. Which is 1.82 point seven, 3.4, 3.8 and 3.9. Okay, so from here. This is er a solution. No, B is asking for some naming or nomenclature. So this is called Les. This is a design metrics. Okay, this is an unknown parameter vector, and this is observation. Victor. Okay. Now for see that is to find the solution of this metric to calculate beta one and beta to We have to convert this rectangular metrics into square metrics will multiply by transfers off this on both sides. That is pre multiplying. So we&#x27;ll have 11 to food. 39 4, 16, 5, 25 times. 111 1, 31 to three, 45 and year. 149 16, 25. This will be my way. Have vita and again here we have to write the same metrics. So 112439 for 16, 5, 25 times 1.82 point 73.43 point eight and 3.9. Okay, so now we have to multiply This sorrow multiplied by column. So this is nothing but square addition right. One square plus two squared plus three squared plus horse square plus five square. So that gives us 55 then multiplying This value with this one gives us to 25. And since it&#x27;s symmetric from this value and this value, it will give him so 2 25 year. Then this gives us nine 79 Okay, so this is Bitta and on multiplying this we get us it is 52.1 and 201.5. Now calculate beater. We&#x27;ll take this right inside by three or it doesn&#x27;t go directly. So we have to pre multiplied by these metrics in Waas here we&#x27;re left with I And here we get 55 to 25 to 25 9 79 raised to minus one. That is, in worse, 52.1 201.5 Seen worse is easy to calculate. So determinant We have to divide by determining which is 55 9, 79 minus 2, 25 square Right. And we have to in touch in these positions in 1979 55 we have to change the sign off these tools Terms of minus 2. 25 minus 2. 25 times 52.1201 point five So on solving this we get Peter is equals to. It&#x27;s just 1.76 and minus zero point 198 So therefore, our equation, why or function? Why is 176 times minus 0.998 x squared? And here we have EC states. That is the answer.

Angelo Rendina · Answer

we want to find their aggression cubic line given by this equation, why equal to be Tawan X plus beater to X square plus beat up three x cubed and were given some data Explain where one x and y n So again, this is an issue off the square solution where we have our constant metrics which is given by the wise off the points. So I want all the way down to y n Want these to be called to well, the system where the valuables are going to be be one bitter too and be the three and the metrics off coefficients he&#x27;s given by Yeah, you see from the equation in the top left that we need to have a better one that multiplies the X. So in the first line we have x one, then beat her to multiplies X squared. So we put X one squared And on the third entry, we have be the three multiplying Excuse. So x one cube it is for the first line and then we repeat these all the way down to x and X and square decks and cubed and these multiplies again. The vector hoodie unknowns better one bitter to embed a three. So we need to solve these problem off the squares. Let&#x27;s write the medics on the left. Little why the matrix off coefficients, Big axe and be done for the valuables. So now we are given this data in the exercise. So I&#x27;ve just written here x and Y, and those are the data they were given. So we need to write the normally questions for it. So I&#x27;ve done it separately. But again, it&#x27;s just a niche of computation. We compute x transpose X distance out to be a three by three metrics with these entries and then x transpose y He&#x27;s a three by one column vector with these entries. And I was just an issue off solving this system So you can do this by Goshen elimination or by dividing well, mostly planned by the inverse of X transpose X. Anyway, you decide to do it. It turns out that the Beetle they were looking for is given by 0.51 minus 0.3 and 0.1 roughly So now that now that we have the coefficients off this cubic line, we can just plot the data. So here I explode to the data there were given that and you see that it&#x27;s kind of our cure Cube only looks like a cubic then. I&#x27;ve also had the program plot to the line with these confusion beaten. So the cubic with the parameters beat up even here by these solution. And it&#x27;s in green. So you see that it&#x27;s a very good fit. Well, a bit last, So going to the right side, but again, easily square solutions. So we&#x27;re minimizing the air or we&#x27;re not finding the exact line, and I would say these works well.

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

Ankit Gupta · Answer

so Hellestrae now and invest in this course in Was Fast Equis and Literacy. The question given a wide equals do 0.68 plus 2.7 and no and the again away. It is given geo point toward one X plus 2.7 on why he Quito. It is given minus You don&#x27;t find six school X plus, then point Geo are it is given judo point 973 hard equal toe Mina Cheeto, point 986 on our equal toe Jiro 0.684 Not unless Chua&#x27;s that aggressing for our say, this is a question one cool 30 and this is giving no now for five and six fate. No wonder to see the father forced to get off Question one. What are the best for food for? It so hosted its location to get you so find equation for the first. Why equal toe minus 0.62 x plus Then why? Because location to get you now the if we&#x27;re able to object, The points are very close to the line. Therefore the best pit are for this. This is equality judo point. My necesito point 98 six for the second second. Good off, but the second good off. I just find the slope. So this Y equal to 0.41 x plus 2.7. It&#x27;s look like this lengths location. Lesser than thought. So that&#x27;s cited in this on DA our artists was U two and the points that are very close to the line, so it minutes are equal to 0.9. So 13 and art is really closed toe one. Then the line Prigent approximately correct part of the doctor and put for the 30. You&#x27;ll find that on the points that very part of a from the given line. So line equation. So why Equal toe slope is more than a second? So why I could Europe on 68 means flushed one x plus 2.7 and are part of this are the points that skitter away from the line. So this is zero point six goal. Four on did answer. The first on the B. The correlation, the correlation coefficient close one is best. What given Doctor

Charles Carter · Answer

Okay, so this phone&#x27;s got several parts. Essay for her. A wants us to solve this system of equations looks rather difficult, but there&#x27;s three point of B uh plus 3.7 a is equal to 105 you know. And for the second Barbie, up 3.7 b plus 4.69 a equaling 1 23.9 And it wants us to solve this because it says in solving it will give us what we need for a linear regression model of a X plus. Be to find a lot of best fit for the data given the table eso. In order to solve this, I multiply my top equation. I think by 3.7 bottom equation by three. You know, Mrs to get my coefficients equal on the values and then we&#x27;ll just subtract the equations so we&#x27;ll find very well have 11.1 b. I have 13.69 a and that&#x27;s gonna be equal. Thio 3 88.5 If now it&#x27;s well spent about 3.7 on the top equation Now the bottom equation Same thing gonna most about by three this time so have 11 by one. Be sticking to me like Arby&#x27;s. We&#x27;re gonna cancel out Game must climb About three is gonna give us 14.7 Just give us a negative slow. Which is what you would expect we can A table like that where the X is going out while the wind is going down, it&#x27;s gonna give us 3 71 0.7. So if we subtract those Oh, let&#x27;s see, we&#x27;re going to have negative 0.38 a equals 16.8 Divide both sides of a negative 0.38 and well, we will have pay is equal to negative 44 0.2105 77 Non repeating decimal. Um, actually, I guess it it will repeat at some point or it&#x27;ll end at some point just because, um, this is a rational number. I said is going to be equal. Negative 44 121 or so. Now, if you play that back in, uh, to our form of sulfur, be waken get 3.70 but 4.69 and synthetic conclave&#x27;s gonna store that number in for a So this is just mine. Answer to the last. I think the only calculator he goes 1 23.9 Amusing the second equation for this. Okay, plugging that in subtracting from both sides. Uh, it&#x27;s gonna give me that be dividing about 3.7. It&#x27;s gonna be 89.5 to 6 and so on and so on. Okay, so it&#x27;s our partner S O A. Is negative. 44.21 be 89.53 Okay, For the second part, it says to plug this into a linear regression feature. So I have my l one and l two. Ah, so playing those in? It&#x27;s just those three values, Madeline Rag of list. Wanless too. For the experts being ling Greg Lynn rag. It&#x27;s usually how it isn&#x27;t calculators listed. So it&#x27;s gonna give me that my A about you. Okay, this should just confirm. So we have negative 44.21 and ours was 215 So this seems to be right and then be is 89.53 and artist was 89.5 to 6. So this does using that graphic Catholic does check now, Part C just says to play it in, so I can&#x27;t I don&#x27;t have quite the graphing abilities on here. Um, but it should look similar to just a line. I was three points around it. Like wine. The best fit style, the way you do it, Like a T I calculator is plug in your X plus b into Why one. Um, sometimes it all automatically saved that in for you when you do your linear regression. And if you look above Why, Why you don&#x27;t have like, uh, plot on? Yes. If he if he had the up arrow and go to that plot, you can turn it on and plot x and y as it is with list one endless two. And then that way you can see that it is a good model. Okay, so for part d, it basically wants us to plug in 1.75 for X. Okay, so when we do that, we plug in a dollar 75. That our demand, or why you is gonna be for 12 0.16 to 5 units. Um, and since we don&#x27;t really talk about units in terms of decimals, I would just say that the demand at a dollar 75 is probably gonna be around 12 units. And this is playing in 1.75 If you stop the graph up, you can do trace and then type in 1.75 and hit. Enter, and I&#x27;ll give you your accident. Live values for that. Okay, Hopefully you learned a few things through here. So part A, we saw for A and B using the equations. Given part B, we plugged in our list, one less to Nis. Linear regression to confirm art theory. Grafted using our Why one under the graphing calculator and 20 our plots for list. One list too. And bar before we plugged in a dollar 75. To get that at a dog, 75 of our demand would be 12 units. Okay, thank you very much.

Jacquelyn Trost · Answer

We&#x27;re going to be finding the best fit line for a set of three points. 12 35 and 46 And we&#x27;re going to do this using the least squares condition. What that means is I&#x27;m going to look at the vertical distance between the points and the best fit line. I&#x27;m going to square those differences and add them together. And then I want to minimize that function. So I&#x27;m gonna be looking at three points on my best fit line x one y one x two y two and x three y three These are the points that are vertically closest to the three points that I&#x27;ve been given, and I&#x27;m going to compare the distance between them now because they are a vertical distance. It means that my ex value is constant in these points. So I know what x one x two and x three are. They match the X values that I was given for the other points. So I&#x27;m just looking at the difference in the wise. So the 1st 1 will be why sub one minus two and I square that distance by sub two minus five squared And why sub three minus six squared. Now I want to get all of this in terms of M and B. Remember, why equals M X plus B? So instead of why for fly sub one, I could have that b m times x sub one or which is one plus B minus two squared for my 2nd 1 Why sub two is going to be three EC. I&#x27;m sorry. Three m plus B minus five squared, and my third point is going to be four m plus B minus six squared. So this is the function I will be using to find the best fit line to find the M and the B for that line. Since we want to minimize the distance, we need to find a critical point. So let&#x27;s start by finding the partial derivative with respect to M of my point. Well, that gives me too. Times M plus B minus two plus two times three m plus B minus five times three plus two times four m plus B minus six times four. And if I simplify that as much as I possibly can, what I end up with is to times 26 m plus eight B minus 41. Now let&#x27;s do the same thing for this partial derivative with respect to be, Hey, that&#x27;s going to give me too. Times M plus B minus two plus two times three m plus B minus five plus two times four m plus B minus six And if I simplify that as much as possible, I will end up with two times AM plus three B minus 13. So I have two equations and I need to set both of these equal to zero to find the critical point. So I have 26 m plus eight B, which would need equal 41 and AM plus three B, which would have to equal 13. Now, if I solve that simultaneous system of equations, I end up with a value of M, which equals 19 14th so and a value of B, which equals 5/7. So my best fit line is M X plus B. Hey, to see what that looks like, you can graph it here. You can see all three of the data points that were given 12 35 and 46 plus the best fit line that we came up with and you can see how good of a fit that line truly is.

Problem

A simple curve that often makes a good model for …

Problem 7 Easy Difficulty

Answer

$y=1.76 x-0.20 x^{2}$

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

Linear Algebra and Its Applications

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$y=1.76 x-0.20 x^{2}$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

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.

Anna Marie V.

Samuel H.

Joseph L.

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

Linear Algebra and Its Applications