Show work by hand for the following please, and will give thumbs up

Post by **answerhappygod** » Wed Jul 06, 2022 12:14 pm

Show work by hand for the following please, and will give thumbsup

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1. [50 points] An article in Optical Engineering, "Operating curve extraction of a correlator's filter" (2004, Vol. 43, pp. 2775-2779) reported on use of an optical correlator to perform an experiment by varying brightness (x₁) and contrast (x₂). The resulting modulation is characterized by the useful range (y) of gray levels. The data are shown below: Brightness (%): Contrast (%): Useful range (ng): 1 X'X= 54 1 61 L56 80 [C (X'X)-¹ = C LC [ 1 X'y = 54 L56 1 61 80 *** (a) [15 points] Fit a multiple linear regression model to these data. Round your answers to 4 decimal places. *** ... 54 61 56 96 *** 1 54 26 1 50 ) ( ) ( ) ( 54 26] TC B₁ = (X'X)-¹X'y = ( 65 96 50 [3]- L255, 50 [1 54 56] 1 61 80 I ⠀ ⠀ [1 54 26] II # 31 100 100 100 65 96 = ) ( ) ( ) ( 50 57 54 25 26 80 155 144 255 ) ( ) ( ) ( 31 Therefore, the multiple linear regression model to these data is ŷ= Bo + B₁x₁ + B₂x₂ = ( ) + ( )x₁ + ( ) ( ) ( )xz 31
(b) [10 points] Estimate o² and the standard errors of the regression coefficients. Round your answers to 3 decimal places. y'y-B'X'y = ( ) SSE n-p se(Bo) = √6² Coo = ( se(B₁)=√√8²C₁1 = ( se (B₂) = √8²C₂z = ( 8² = = y'y - B'X'y n-p Fo SSR = B'X'y- MSR MSE (c) [2 points] Use the model developed in (a) to predict Useful Range when x₁ = 80 and X₂ = 75. Round your answer to 3 decimal places. Đ = Bot Box+ B2xz = Bo + Bi (80) + $,(75) = ( 1 (d) [8 points] Test for significance of regression using a = 0.05. Round your answers to 3 decimal places. Ho: B₁ B₂ = 0 ( (Ey₁)² n = ( = ( 1 161.4 199.5 215.7 2 18.51 19.00 19.16 3 10.13 9.55 9.28 7.71 6.59 4 5 6.94 6.61 5.79 5.41 6 5.99 5.14 4.76 7 5.59 4.74 4.35 ) ) ) ) ) 4.53 4.12 Since fo is ( ) than fa,k,n-p = ( hypothesis and conclude that the regression model ( for ) H₁: B, 0 for at least one j ) ), we should ( 7 224.6 230.2 234.0 236.8 19.25 19.30 19.33 19.35 9.12 9.01 6.39 6.26 8.94 6,16 8.89 8.85 6.04 4.82 6.09 5.19 5.05 4.95 4,88 4.39 4.28 4.21 4.15 3.97 3.87 3.79 3.73 ) the null ) significant at a = 0.05. Degrees of freedom for the numerator (₁) 8 10 238.9 240.5 241.9 19.37 19.38 19.40 8.81 8.79 6.00 5.96 4.77 4.74 12 15 20 243.9 245.9 248.0 19.41 19.43 19.45 8.74 8.70 8.66 5.91 5.86 5.80 4.68 4.62 4.56 4.10 4.06 4.00 3.94 3.87 3.68 3.64 3.57 3.51 3.44
(e) [15 points] Construct a t-test on each regression coefficient. What conclusions can you draw about the variables in this model? Use a = 0.05. Round your answers to 3 decimal places. For the coefficient B₁, Ho: B₁ = 0 H₁: B₁ * 0 B₁-B10 B₁ Since |tol = -₁ == C ( √0²C₁1 se(B₁ we should ( For the coefficient B2, Ho: B₂ = 0 H₁: B₂0 B₂-B₂0 Since |to|=|-|=||= C ( we should ( Conclusion: 1 2 3 4 5 6 Percentage Points ta,e of the t Distribution degrees of freedom. 7 8 9 10 11 12 13 14 15 40 25 325 1.000 ) the null hypothesis for the coefficient ₁. B₂ 289 277 271 267 265 263 262 261 260 ) is ( .10 3.078 ) is ( ) than ta/2,n-p = ( ) the null hypothesis for the coefficient $₂. .05 025 6.314 12.706 2.920 4.303 2.353 3.182 2.132 2.776 2.015 2.571 1.943 2.447 2.365 2.306 .01 .005 .0025 .001 31.821 63.657 127.32 318.31 6.965 9.925 14.089 23.326 816 1.886 7765 1.638 4.541 5.841 7.453 10.213 .741 1.533 3,747 4.604 5.598 7.173 727 1.476 3.365 4.032 .718 1.440 3.143 3.707 711 1.415 1.895 2.998 3.499 4.029 .706 1.397 1.860 2.896 3.355 3.833 703 1.383 1.833 2.262 2.821 3.250 3.690 .700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 260 .697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 259 .695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 259 694 1.350 1.771 2.160 2.650 258 692 1.345 1.761 2.145 258 .691 1.341 1.753 2.131 2.602 ) than ta/2.n-p = ( 3.012 2.624 2.977 4,773 5.893 4.317 5.208 4.785 4.501 4.297 2.947 3.286 0005 636.62 31.598 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 3.372 3.852 4.221 3.326 3.787 4.140 3.733 4.073 ), ),