Interpretation:
The graph shows an increasing trend and a positive association between the size of the diamond (carat) and the cost (in US dollars). This means that as the size of the diamond (carat) increases, the cost of the diamond will also increase.
Linear Model:
Y=8225.1x-558.52
The linear model given above can help us predict the cost of the diamond.
- To check if the linear model is appropriate, it needs to fulfil three conditions: linearity, nearly normal residuals, and constant variability. The data set given in this question fulfils these conditions.
- The slope of this model is 8225.1. It helps estimate the average change in the dependent variable (cost) for each unit change in the independent variable (size).
- The intercept of this model is -558.52, which helps measure the variability in the response when the independent variable is equal to zero.
- As the Coefficient of Determination is r^2 = .9875, after taking the square root, we shall get the value of r=.99373. The value of the correlation coefficient r is closer to 1, which means that there is a strong positive correlation between the two variables.
- The cost of a quarter of a diamond and the residual values are given below:
Working
- Firstly, the dependent variable (the size of the diamond), each value, was divided into four columns to get the required column. The linear equation was calculated again. After putting the values of column Quarter of the diamond in the regression equation and solving it, we obtained the Y1^. Finally, by subtracting the observed value (Y) from the estimating value (Y1^) we get the value of residual error.
| Cost of Diamonds ($) | Size of Diamonds | y^ | Residual (Y-Y^) | A quarter of diamond (x1) | Revised Cost Y1^ | Residual (y-y^) |
| 494.82 | 0.12 | 428.49 | 66.33 | 0.03 | 428.48 | 66.34 |
| 768.03 | 0.17 | 839.75 | -71.72 | 0.0425 | 839.73 | -71.70 |
| 1105.03 | 0.2 | 1086.50 | 18.53 | 0.05 | 1086.48 | 18.55 |
| 1508.88 | 0.25 | 1497.76 | 11.13 | 0.0625 | 1497.73 | 11.15 |
| 1826.18 | 0.28 | 1744.51 | 81.67 | 0.07 | 1744.48 | 81.70 |
| 2096.89 | 0.33 | 2155.76 | -58.87 | 0.0825 | 2155.73 | -58.84 |
| 688.24 | 0.15 | 675.25 | 12.99 | 0.0375 | 675.23 | 13.01 |
| 944.9 | 0.18 | 922.00 | 22.90 | 0.045 | 921.98 | 22.92 |
| 1071.75 | 0.21 | 1168.75 | -97.00 | 0.0525 | 1168.73 | -96.98 |
| 1504.44 | 0.26 | 1580.01 | -75.57 | 0.065 | 1579.98 | -75.54 |
| 1908.28 | 0.29 | 1826.76 | 81.52 | 0.0725 | 1826.73 | 81.55 |
| 2409.76 | 0.35 | 2320.27 | 89.50 | 0.0875 | 2320.23 | 89.53 |
| 748.1 | 0.16 | 757.50 | -9.40 | 0.04 | 757.48 | -9.38 |
| 1076.18 | 0.19 | 1004.25 | 71.93 | 0.0475 | 1004.23 | 71.95 |
| 1289.2 | 0.23 | 1333.25 | -44.05 | 0.0575 | 1333.23 | -44.03 |
| 1597.63 | 0.27 | 1662.26 | -64.63 | 0.0675 | 1662.23 | -64.60 |
| 2038.09 | 0.32 | 2073.51 | -35.42 | 0.08 | 2073.48 | -35.39 |
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