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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.

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

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 6

Applications to Linear Models

Vectors

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Okay, so questioning certain experiment produces the data is given. I'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's say four x one y one pair, we'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'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'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'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'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'll take this right inside by three or it doesn't go directly. So we have to pre multiplied by these metrics in Waas here we'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'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.

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