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Homework 5 Solutions Ch. 8 2,4,5,10,18,23,24 8.2 ∑ x ∑ y − ∑ x ∑ xy = 20 ⋅ 24 − 0 ⋅ 22 = 6 4 ⋅ 20 − 0 N ∑ x − (∑ x ) N ∑ xy − ∑ x ∑ y 4 ⋅ 22 − 0 ⋅ 24 = = 1 .1 B= 80 N ∑ x − (∑ x ) 2 A= 2 2 2 2 2 Therefore, the equation for the line is y = 1.1x + 6.0 8.4 From equation 8.8, AN + B ∑ x = ∑ y Divide by N: ∑ x = ∑ y or A+ B A + Bx = y N N Therefore, the point (x, y ) lies on the best-fit line. 8.5 For a line through the origin, A = 0, so χ 2 = ∑ Minimize this with respect to B: Therefore, ∑x y i i ( yi − Bxi )2 σ y2 ∂χ 2 − 2 = ∑ xi ( yi − Bxi ) = 0 ∂B σ y2 − B ∑ xi2 = 0 or B = ∑ xy ∑x 2 8.10 For the weighted fit, the weights are inversely proportional to the uncertainies squared: wi = (1 / 0.5 2 ,1 / 0.5 2 ,1 / 12 ) = (4,4,1) Using these weights, calculate A and B: ∑ wx 2 ∑ wy − ∑ wx ∑ wxy = 29 ⋅ 22 − 15 ⋅ 38 = 1.89 A= 2 9 ⋅ 29 − 15 2 ∑ w∑ wx 2 − (∑ wx ) B= ∑ w∑ wxy − ∑ wx ∑ wy = 9 ⋅ 38 − 15 ⋅ 22 = 0.33 36 ∑ w∑ wx − (∑ wx ) 2 2 Now calculate A and B for unweighted data points: ∑ x 2 ∑ y − ∑ x ∑ xy = 14 ⋅ 7 − 6 ⋅14 = 2.33 A= 2 3 ⋅14 − 6 2 N ∑ x 2 − (∑ x ) B= N ∑ xy − ∑ x ∑ y N ∑ x 2 − (∑ x ) 2 = 3 ⋅14 − 6 ⋅ 7 = 0.00 6 8.18 We have B = ∑ xy , where ∆ does not depend on y. ∆ we need its derivatives with respect to the y’s: ∂B xi = ∂y i ∆ ∑ (x σ ) = 2 Therefore, σ 2 B i y 2 ∆ ∑x (∑ x ) 2 =σ 2 y 2 2 = σ y2 ∑x 2 To calculate the uncertainty in B, σB = σy ∑x 2 8.23 We have χ 2 = ∑ ( Af i + Bg i − yi ) / σ y2 , where f i ≡ f (xi ) and g i ≡ g (xi ) 2 Minimize this with respect to A and B: ∂χ 2 ∂χ 2 ∝ ∑ f i (Af i + Bg i − yi ) = 0 ∝ ∑ g i ( Af i + Bg i − yi ) = 0 ∂B ∂A Simplify this a bit: A∑ f 2 + B ∑ fg = ∑ fy A∑ fg + B ∑ g 2 = ∑ gy These are the equations given. 8.24 We will use the equations from the problem above. First, calculate the function at the various values of x and use this to calculate the coefficients in the equations. The results are: ∑ f 2 = 2.22 ∑ g 2 = 2.78 ∑ fg = 0.00 ∑ fy = 6.48 ∑ gy = 14.63 The equations become 2.22 A = 6.48 2.78 B = 14.63 Therefore, A = 2.92 B = 5.26 The fit is rather poor – it is likely the frequency is not correct.