1. **Problem Statement:** We are given data on inspection expenditure and defective parts per 1000 units for 3 plants. We need to find the Coefficient of Determination ($R^2$) and the Coefficient of Correlation ($r$). 2. **Data:** Inspection expenditure ($x$): 27, 33, 17 Defective parts ($y$): 53, 36, 63 3. **Formulas:** - Mean of $x$: $\bar{x} = \frac{\sum x_i}{n}$ - Mean of $y$: $\bar{y} = \frac{\sum y_i}{n}$ - Covariance: $\mathrm{cov}(x,y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n}$ - Variance: $\mathrm{var}(x) = \frac{\sum (x_i - \bar{x})^2}{n}$, similarly for $y$ - Correlation coefficient: $r = \frac{\mathrm{cov}(x,y)}{\sqrt{\mathrm{var}(x) \cdot \mathrm{var}(y)}}$ - Coefficient of Determination: $R^2 = r^2$ 4. **Calculate means:** $$\bar{x} = \frac{27 + 33 + 17}{3} = \frac{77}{3} = 25.67$$ $$\bar{y} = \frac{53 + 36 + 63}{3} = \frac{152}{3} = 50.67$$ 5. **Calculate deviations and products:** | $x_i$ | $y_i$ | $x_i - \bar{x}$ | $y_i - \bar{y}$ | $(x_i - \bar{x})(y_i - \bar{y})$ | $(x_i - \bar{x})^2$ | $(y_i - \bar{y})^2$ | |-------|-------|-----------------|-----------------|-------------------------------|-------------------|-------------------| | 27 | 53 | 1.33 | 2.33 | 3.10 | 1.77 | 5.43 | | 33 | 36 | 7.33 | -14.67 | -107.56 | 53.74 | 215.17 | | 17 | 63 | -8.67 | 12.33 | -106.89 | 75.17 | 152.06 | 6. **Sum values:** $$\sum (x_i - \bar{x})(y_i - \bar{y}) = 3.10 - 107.56 - 106.89 = -211.35$$ $$\sum (x_i - \bar{x})^2 = 1.77 + 53.74 + 75.17 = 130.68$$ $$\sum (y_i - \bar{y})^2 = 5.43 + 215.17 + 152.06 = 372.66$$ 7. **Calculate covariance and variances:** $$\mathrm{cov}(x,y) = \frac{-211.35}{3} = -70.45$$ $$\mathrm{var}(x) = \frac{130.68}{3} = 43.56$$ $$\mathrm{var}(y) = \frac{372.66}{3} = 124.22$$ 8. **Calculate correlation coefficient:** $$r = \frac{-70.45}{\sqrt{43.56 \times 124.22}} = \frac{-70.45}{\sqrt{5410.5}} = \frac{-70.45}{73.58} = -0.958$$ 9. **Calculate coefficient of determination:** $$R^2 = (-0.958)^2 = 0.918$$ **Final answers:** - Coefficient of Determination ($R^2$) = 0.918 - Coefficient of Correlation ($r$) = -0.958