The price (in $) and the carat weight of a diamond are its two most known charac

Question

The price (in $) and the carat weight of a diamond are its two most known characteristics. In order to understand the role carat weight has in determioning the price of a diamond, the carat weight and the price of 20 round diamonds, all of color D and clarity VS1, were obtained recently as shown below.

Carat Weight   & Price

0.56          2789

0.60          2517

0.56          2645

0.57          2367

0.53          2673

0.61          3029

0.53          2701

0.51          2549

0.67         2959

0.54         2276

0.59         2841

0.65         2853

0.51          2024

0.54          2609

0.51           2603

0.51           2159

0.57           2398

0.52         2061

0.56         2328

0.51          2047

Question # 1    Use the computer to find the equation for the line of best fit.

Question # 2    On the average, by how much does the price increase for each extre 0.01 carat weight? Within what interval of x-values would you expect this to be true?

Question # 3 Use the computer to find the variance of y about the regression line.

Explanation / Answer

from above

1) b1=Sxy/Sxx=4356.88

and bo=Ybar-b1*Xbar =92.44

hence euqation for line of best fit : y=4356.88x+92.44

2) for each extre 0.01 carat weight price increase =0.01*4356.88=43.57

we expect this to be true with in given range of values from 0.51 carat to 0.67 caarat.

3) from above variance of Y about the regression line =SSE/(n-2) =54120.15

s. No x Y (X-Xbar)^2 (Y-Ybar)^2 (X-Xbar)(Y-Ybar) Yi'=b0+x*b1 (Y-Yi')^2 (Ybar-Yi')^2 1 0.56 2789 0.0000 71609.760 0.669 2532.2922 65898.8957 118.64 2 0.6 2517 0.0018 19.360 -0.187 2706.5674 35935.7846 34286.95 3 0.56 2645 0.0000 15276.960 0.309 2532.2922 12703.0487 118.64 4 0.57 2367 0.0002 23839.360 -1.930 2575.8610 43622.9126 2966.00 5 0.53 2673 0.0008 22982.560 -4.169 2401.5858 73665.6544 14355.44 6 0.61 3029 0.0028 257657.760 26.649 2750.1362 77765.0455 52320.23 7 0.53 2701 0.0008 32256.160 -4.939 2401.5858 89648.8483 14355.44 8 0.51 2549 0.0023 761.760 -1.311 2314.4482 55014.5267 42829.03 9 0.67 2959 0.0127 191493.760 49.230 3011.5489 2761.3867 240245.94 10 0.54 2276 0.0003 60221.160 4.294 2445.1546 28613.2841 5813.36 11 0.59 2841 0.0011 102144.160 10.387 2662.9986 31684.5089 20050.16 12 0.65 2853 0.0086 109958.560 30.673 2924.4113 5099.5761 162418.12 13 0.51 2024 0.0023 247406.760 23.627 2314.4482 84360.1819 42829.03 14 0.54 2609 0.0003 7673.760 -1.533 2445.1546 26845.3099 5813.36 15 0.51 2603 0.0023 6658.560 -3.876 2314.4482 83262.1165 42829.03 16 0.51 2159 0.0023 131333.760 17.214 2314.4482 24164.1563 42829.03 17 0.57 2398 0.0002 15227.560 -1.543 2575.8610 31634.5313 2966.00 18 0.52 2061 0.0014 211968.160 17.265 2358.0170 88219.1185 26693.99 19 0.56 2328 0.0000 37403.560 -0.484 2532.2922 41735.3021 118.64 20 0.51 2047 0.0023 225055.360 22.534 2314.4482 71528.5627 42829.03 total(T) 11.1500 50428.0000 0.0420 1770948.8000 182.880 974162.7512 796786.049 mean=T/n 0.558 2521.40 Sxx Syy Sxy

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