1. **Problem Statement:** Given five observations for two variables $x$ and $y$: $x$: 4, 6, 11, 3, 16 $y$: 50, 50, 40, 60, 30 Compute the sample covariance and sample correlation coefficient. 2. **Formulas and Important Rules:** - Sample covariance formula: $$\text{Cov}(x,y) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$$ - Sample correlation coefficient formula: $$r = \frac{\text{Cov}(x,y)}{s_x s_y}$$ where $s_x$ and $s_y$ are the sample standard deviations of $x$ and $y$ respectively. - Mean of $x$: $$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$ - Mean of $y$: $$\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$$ 3. **Calculate Means:** $$\bar{x} = \frac{4 + 6 + 11 + 3 + 16}{5} = \frac{40}{5} = 8$$ $$\bar{y} = \frac{50 + 50 + 40 + 60 + 30}{5} = \frac{230}{5} = 46$$ 4. **Calculate Deviations and Products:** | $x_i$ | $y_i$ | $x_i - \bar{x}$ | $y_i - \bar{y}$ | $(x_i - \bar{x})(y_i - \bar{y})$ | |-------|-------|-----------------|-----------------|-------------------------------| | 4 | 50 | 4 - 8 = -4 | 50 - 46 = 4 | (-4)(4) = -16 | | 6 | 50 | 6 - 8 = -2 | 50 - 46 = 4 | (-2)(4) = -8 | | 11 | 40 | 11 - 8 = 3 | 40 - 46 = -6 | (3)(-6) = -18 | | 3 | 60 | 3 - 8 = -5 | 60 - 46 = 14 | (-5)(14) = -70 | | 16 | 30 | 16 - 8 = 8 | 30 - 46 = -16 | (8)(-16) = -128 | Sum of products: $$-16 - 8 - 18 - 70 - 128 = -240$$ 5. **Calculate Sample Covariance:** $$\text{Cov}(x,y) = \frac{-240}{5 - 1} = \frac{-240}{4} = -60$$ 6. **Calculate Sample Standard Deviations:** Calculate $s_x$: $$\sum (x_i - \bar{x})^2 = (-4)^2 + (-2)^2 + 3^2 + (-5)^2 + 8^2 = 16 + 4 + 9 + 25 + 64 = 118$$ $$s_x = \sqrt{\frac{118}{4}} = \sqrt{29.5} \approx 5.431$$ Calculate $s_y$: $$\sum (y_i - \bar{y})^2 = 4^2 + 4^2 + (-6)^2 + 14^2 + (-16)^2 = 16 + 16 + 36 + 196 + 256 = 520$$ $$s_y = \sqrt{\frac{520}{4}} = \sqrt{130} \approx 11.402$$ 7. **Calculate Sample Correlation Coefficient:** $$r = \frac{-60}{5.431 \times 11.402} = \frac{-60}{61.91} \approx -0.969$$ **Interpretation:** - The sample covariance is negative, indicating that as $x$ increases, $y$ tends to decrease. - The correlation coefficient is approximately $-0.969$, which is very close to $-1$, indicating a strong negative linear relationship between $x$ and $y$. **Final Answers:** Sample covariance = $-60$ Sample correlation coefficient = $-0.969$