Find the equation y = a0 + a1x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).

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Suresh Chand · Accepted Answer

Solution: Given, y = a0 + a1x  --------- (i) Then at the points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2), the line (i) gives, (left.begin{matrix}1 = y_1 = a_0 1 = y_2 = a_0 + a_2 1 = y_3 = a_0 + a_2 2 = y_4 = a_0 + 2a_2 2 = y_5 = a_0 + 3a_2 end{matrix}right})   --------- (ii) The matrix form of equation (ii) is, y = ax ----------- (iii) where, y = (begin{bmatrix}y_1y_2y_3y_4y_5end{bmatrix}) = (begin{bmatrix}11122end{bmatrix}) X = (begin{bmatrix}1 & 0 1 & 1 1 & 1 1 & 2 1 & 3end{bmatrix}) a = (begin{bmatrix}a_0 a_1end{bmatrix}) The normal equation to (iii) is, XTX a = XTy    --------- (iv) Here, XTX = (begin{bmatrix}1 & 1 & 1 & 1 & 1 0 & 1 & 1 & 2 & 3end{bmatrix}begin{bmatrix}1 & 0 1 & 1 1 & 1 1 & 2 1 & 3end{bmatrix}) = (begin{bmatrix}5 & 7 7 & 15end{bmatrix}) and, XTy = (begin{bmatrix}1 & 1 & 1 & 1 & 1 0 & 1 & 1 & 2 & 3end{bmatrix} begin{bmatrix}11122end{bmatrix}) = (begin{bmatrix}7 12end{bmatrix}) Therefore, (iv) becomes, (begin{bmatrix}5 & 7 7 & 15end{bmatrix})(begin{bmatrix}a_0 a_1end{bmatrix}) = (begin{bmatrix}7 12end{bmatrix}) (begin{bmatrix}5a_0 + 7a_1 7a_0 + 15a_1end{bmatrix}) = (begin{bmatrix}7 12end{bmatrix}) Solving this, we get a0 = 21/26 a1 = 11/26 Thus, Equation (i) becomes, y = (frac{21}{26} + frac{11x}{26}) This is the required least square line.

Find the equation y = a₀ + a₁x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).

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Find the equation y = a0 + a1x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).

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Find the equation y = a₀ + a₁x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).