A certain experiment produces the data (1, 1.6), (2, 2.8). (3, 3.2), (4, 3.8), a

Question

A certain experiment produces the data (1, 1.6), (2, 2.8). (3, 3.2), (4, 3.8), and (5, 4.1). Describe the model that produces a least-squares fit of these points by a function of the form y-81 x + 2xf. 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 set for the product. a. Give the design matrix and observation vector for the unknown parameter vector = b. Find the associated least-squares curve for the data a. The design matrix is X= | 3 9 4 16 5 25 1.6 2.8 | 3.2 3.8 4.1 The observation vector is y b. The least-squares curve for the data is given by the function y= (Round to two decimal places as needed.)

Explanation / Answer

GIVEN FUNCTION IS OF PAROBOLIC FORM AS IT IS Y = 1X + 2 X2

FOR PARABOLIC MODEL WE USE bY = a + b x + c x3 but in this case a = 0

solving a parabolic least sqare fit we use three normal equation as

Y= na + b X + c x2 ,  XY = a  x + b  X2 + c X3, , X2Y = a X2 + b X3 + c  x4 BUT IN THIS MODEL

a = 0 ,hence our required normal equations will be

Y= b X + c x2 and   XY = b  X2 + c X3  ,    NO NEED OF THIRD AS LEFT WITH TWO VARIABLLES

HENCE WE HAVE TO CALCULATE 12 WITH HELP OF   Y= b X + c x2 AND XY = b X2 + c X3  WHERE b =  1, and c=2

TABLE FOR CALCULATION OF  1 , 2

X Y XY X2 X2Y X3

1 1.6 1.6 1 1.6 1

2 2.8 5.6 4 11.2 8

3 3.2 9.6 9 28.8 27

4 3.8 15.2 16 60.8 64

5 4.1 20.5 25 102.5 125

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15 15.5 52.5 55 204.9 225

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NOW    X = 15 ,   Y= 15.5 ,   XY = 52.5 , X2 = 55 AND    X2Y = 204.9

using two normal equations 15.5 = 15 b + 55 c

and 52.5 = 55b + 225 c solving these equations we find b = 12/7 =  1, and c = -13/70=2

HENCE REQUIRED MODEL IS  Y = 12/7 X + (- 13/70 ) X2

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