1. **State the problem:** We need to find the coefficient of determination, $R^2$, for the given data points $(x, y)$. The coefficient of determination measures how well the regression line fits the data. 2. **Given data:** $$x = [2500, 0.1346, 5000, 1250, 1700, 1937, 2200, 2249, 2500]$$ $$y = [458.5, 447.0, 695.0, 114.9, 158.8, 349.9, 369.0, 425.0, 469.0]$$ 3. **Calculate means:** $$\bar{x} = \frac{\sum x_i}{n} = \frac{2500 + 0.1346 + 5000 + 1250 + 1700 + 1937 + 2200 + 2249 + 2500}{9}$$ Calculate sum: $$2500 + 0.1346 + 5000 + 1250 + 1700 + 1937 + 2200 + 2249 + 2500 = 19336.1346$$ So, $$\bar{x} = \frac{19336.1346}{9} \approx 2148.46$$ Similarly for $y$: $$\bar{y} = \frac{458.5 + 447.0 + 695.0 + 114.9 + 158.8 + 349.9 + 369.0 + 425.0 + 469.0}{9}$$ Sum: $$458.5 + 447.0 + 695.0 + 114.9 + 158.8 + 349.9 + 369.0 + 425.0 + 469.0 = 3487.1$$ So, $$\bar{y} = \frac{3487.1}{9} \approx 387.46$$ 4. **Calculate sums for regression coefficients:** $$S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})$$ $$S_{xx} = \sum (x_i - \bar{x})^2$$ Calculate each term: | $x_i$ | $y_i$ | $x_i - \bar{x}$ | $y_i - \bar{y}$ | $(x_i - \bar{x})(y_i - \bar{y})$ | $(x_i - \bar{x})^2$ | |-------|-------|-----------------|-----------------|-------------------------------|-------------------| | 2500 | 458.5 | 351.54 | 71.04 | 24956.4 | 123580.6 | | 0.1346| 447.0 | -2148.33 | 59.54 | -127865.3 | 4618257.5 | | 5000 | 695.0 | 2851.54 | 307.54 | 877134.3 | 8133817.6 | | 1250 | 114.9 | -898.46 | -272.56 | 244812.3 | 807230.3 | | 1700 | 158.8 | -448.46 | -228.66 | 102535.3 | 201120.3 | | 1937 | 349.9 | -211.46 | -37.56 | 7943.3 | 44715.3 | | 2200 | 369.0 | 51.54 | -18.46 | -951.7 | 2656.3 | | 2249 | 425.0 | 100.54 | 37.54 | 3773.3 | 10108.3 | | 2500 | 469.0 | 351.54 | 81.54 | 28670.3 | 123580.6 | Sum these: $$S_{xy} = 24956.4 - 127865.3 + 877134.3 + 244812.3 + 102535.3 + 7943.3 - 951.7 + 3773.3 + 28670.3 = 1,217,107.9$$ $$S_{xx} = 123580.6 + 4618257.5 + 8133817.6 + 807230.3 + 201120.3 + 44715.3 + 2656.3 + 10108.3 + 123580.6 = 13,628,966.8$$ 5. **Calculate slope $b$ and intercept $a$ of regression line:** $$b = \frac{S_{xy}}{S_{xx}} = \frac{1,217,107.9}{13,628,966.8} \approx 0.0893$$ $$a = \bar{y} - b \bar{x} = 387.46 - 0.0893 \times 2148.46 \approx 387.46 - 191.87 = 195.59$$ 6. **Calculate total sum of squares $SST$ and regression sum of squares $SSR$:** $$SST = \sum (y_i - \bar{y})^2$$ $$SSR = b \times S_{xy}$$ Calculate $SST$: $$(71.04)^2 + (59.54)^2 + (307.54)^2 + (-272.56)^2 + (-228.66)^2 + (-37.56)^2 + (-18.46)^2 + (37.54)^2 + (81.54)^2 =$$ $$5046.7 + 3545.1 + 94588.7 + 74390.3 + 52292.3 + 1410.7 + 340.8 + 1409.5 + 6650.0 = 235674.1$$ Calculate $SSR$: $$SSR = 0.0893 \times 1,217,107.9 = 108,635.3$$ 7. **Calculate coefficient of determination $R^2$:** $$R^2 = \frac{SSR}{SST} = \frac{108,635.3}{235,674.1} \approx 0.461$$ **Final answer:** The coefficient of determination for the data is approximately $$\boxed{0.461}$$ This means about 46.1% of the variation in $y$ can be explained by the linear relationship with $x$.