1. **Problem 10: Scatter Diagram and Correlation Sign** Given data: Mathematics (X): 15, 18, 21, 24, 27, 30, 36, 39, 42, 48 Statistics (Y): 25, 25, 27, 27, 31, 33, 35, 41, 41, 45 - We plot each pair $(X_i,Y_i)$ on the coordinate plane. - By observing the points, as Mathematics marks increase, Statistics marks also tend to increase. - This indicates a **positive correlation** between Mathematics and Statistics scores. 2. **Problem 11: Find Covariance of X and Y** Given: $X = [1,2,3,4,5]$, $Y = [2,4,6,8,10]$ - Calculate means: $$\bar{X} = \frac{1+2+3+4+5}{5} = 3$$ $$\bar{Y} = \frac{2+4+6+8+10}{5} = 6$$ - Covariance formula: $$\mathrm{Cov}(X,Y) = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$$ - Compute products: \[(1-3)(2-6) = (-2)(-4) = 8\] \[(2-3)(4-6) = (-1)(-2) = 2\] \[(3-3)(6-6) = 0\] \[(4-3)(8-6) = 1 \times 2 = 2\] \[(5-3)(10-6) = 2 \times 4 = 8\] - Sum = $8 + 2 + 0 + 2 + 8 = 20$ - Divide by $n=5$: $$\mathrm{Cov}(X,Y) = \frac{20}{5} = 4$$ 3. **Problem 12: Covariance for new data** Given: $X = [64,65,66,67,68,69,70]$ $Y = [66,67,65,68,70,68,72]$ - Calculate means: $$\bar{X} = \frac{64+65+66+67+68+69+70}{7} = 67$$ $$\bar{Y} = \frac{66+67+65+68+70+68+72}{7} = 67.4286$$ (approx) - Calculate $(X_i - \bar{X})(Y_i - \bar{Y})$ for each pair: \[(64-67)(66-67.4286) = (-3)(-1.4286) = 4.2858\] \[(65-67)(67-67.4286) = (-2)(-0.4286) = 0.8572\] \[(66-67)(65-67.4286) = (-1)(-2.4286) = 2.4286\] \[(67-67)(68-67.4286) = 0 \times 0.5714 = 0\] \[(68-67)(70-67.4286) = 1 \times 2.5714 = 2.5714\] \[(69-67)(68-67.4286) = 2 \times 0.5714 = 1.1428\] \[(70-67)(72-67.4286) = 3 \times 4.5714 = 13.7142\] - Sum these values: $$4.2858 + 0.8572 + 2.4286 + 0 + 2.5714 + 1.1428 + 13.7142 = 24$$ (approx) - Divide by n = 7: $$\mathrm{Cov}(X,Y) \approx \frac{24}{7} = 3.4286$$ 4. **Problem 13: Covariance with given X and Y** Given: $X = [1,2,3,4,5,6,7,8,9,10]$ $Y = [10,9,8,8,6,12,4,3,18,1]$ - Calculate means: $$\bar{X} = \frac{1+...+10}{10} = 5.5$$ $$\bar{Y} = \frac{10+9+8+8+6+12+4+3+18+1}{10} = 7.9$$ - Calculate $(X_i - \bar{X})(Y_i - \bar{Y})$ for each i: \[(1-5.5)(10-7.9) = (-4.5)(2.1) = -9.45\] \[(2-5.5)(9-7.9) = (-3.5)(1.1) = -3.85\] \[(3-5.5)(8-7.9) = (-2.5)(0.1) = -0.25\] \[(4-5.5)(8-7.9) = (-1.5)(0.1) = -0.15\] \[(5-5.5)(6-7.9) = (-0.5)(-1.9) = 0.95\] \[(6-5.5)(12-7.9) = (0.5)(4.1) = 2.05\] \[(7-5.5)(4-7.9) = (1.5)(-3.9) = -5.85\] \[(8-5.5)(3-7.9) = (2.5)(-4.9) = -12.25\] \[(9-5.5)(18-7.9) = (3.5)(10.1) = 35.35\] \[(10-5.5)(1-7.9) = (4.5)(-6.9) = -31.05\] - Sum these: $$-9.45 -3.85 -0.25 -0.15 +0.95 +2.05 -5.85 -12.25 +35.35 -31.05 = -24.45$$ - Divide by $n=10$: $$\mathrm{Cov}(X,Y) = \frac{-24.45}{10} = -2.445$$ **Final answers:** 10. Correlation is **positive**. 11. $\mathrm{Cov}(X,Y) = 4$ 12. $\mathrm{Cov}(X,Y) \approx 3.43$ 13. $\mathrm{Cov}(X,Y) \approx -2.45$