1. Problem 10: Draw a scatter diagram for Mathematics and Statistics marks and determine correlation. - Mathematics marks: $15, 18, 21, 24, 27, 30, 36, 39, 42, 48$ - Statistics marks: $25, 25, 27, 27, 31, 33, 35, 41, 41, 45$ Step 1: Plot each pair $(x,y)$ where $x$ is Mathematics mark and $y$ is Statistics mark. Step 2: Observing the data, as Mathematics marks increase, Statistics marks also increase. Step 3: This indicates a positive correlation. --- 2. Problem 11: Find covariance $Cov(X,Y)$ where $X = 1,2,3,4,5$ and $Y = 2,4,6,8,10$ Step 1: Calculate means: $$\bar{X} = \frac{1+2+3+4+5}{5} = 3$$ $$\bar{Y} = \frac{2+4+6+8+10}{5} = 6$$ Step 2: Calculate deviations product sum: $$\sum (X_i - \bar{X})(Y_i - \bar{Y}) = (1-3)(2-6)+(2-3)(4-6)+(3-3)(6-6)+(4-3)(8-6)+(5-3)(10-6)$$ $$= (-2)(-4)+(-1)(-2)+0+1\times2+2\times4 = 8 + 2 + 0 + 2 + 8 = 20$$ Step 3: Covariance: $$Cov(X,Y) = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{n} = \frac{20}{5} = 4$$ --- 3. Problem 12: Find covariance between $X = 64, 65, 66, 67, 68, 69, 70$ and $Y = 66, 67, 65, 68, 70, 68, 72$ Step 1: Calculate means: $$\bar{X} = \frac{64+65+66+67+68+69+70}{7} = 67$$ $$\bar{Y} = \frac{66+67+65+68+70+68+72}{7} = 67.43$$ Step 2: Calculate $\sum (X_i-\bar{X})(Y_i-\bar{Y})$: Compute each product: $(64-67)(66-67.43) = (-3)(-1.43) = 4.29$ $(65-67)(67-67.43) = (-2)(-0.43) = 0.86$ $(66-67)(65-67.43) = (-1)(-2.43) = 2.43$ $(67-67)(68-67.43) = 0\times0.57 =0$ $(68-67)(70-67.43) = 1 \times 2.57 = 2.57$ $(69-67)(68-67.43) = 2 \times 0.57 = 1.14$ $(70-67)(72-67.43) = 3 \times 4.57 = 13.71$ Sum = $4.29+0.86+2.43+0+2.57+1.14+13.71=24.99$ Step 3: Covariance: $$Cov(X,Y) = \frac{24.99}{7} \approx 3.57$$ --- 4. Problem 13: Find covariance between $X = 1,2,3,4,5,6,7,8,9,10$ $Y = 10,9,8,8,6,12,4,3,18,1$ Step 1: Means: $$\bar{X} = \frac{55}{10} = 5.5$$ $$\bar{Y} = \frac{79}{10} = 7.9$$ Step 2: Compute $\sum (X_i - \bar{X})(Y_i - \bar{Y})$: Computed individually: $(1-5.5)(10-7.9)=(-4.5)(2.1)=-9.45$ $(2-5.5)(9-7.9)=(-3.5)(1.1)=-3.85$ $(3-5.5)(8-7.9)=(-2.5)(0.1)=-0.25$ $(4-5.5)(8-7.9)=(-1.5)(0.1)=-0.15$ $(5-5.5)(6-7.9)=(-0.5)(-1.9)=0.95$ $(6-5.5)(12-7.9)=(0.5)(4.1)=2.05$ $(7-5.5)(4-7.9)=(1.5)(-3.9)=-5.85$ $(8-5.5)(3-7.9)=(2.5)(-4.9)=-12.25$ $(9-5.5)(18-7.9)=(3.5)(10.1)=35.35$ $(10-5.5)(1-7.9)=(4.5)(-6.9)=-31.05$ Sum = $-9.45 -3.85 -0.25 -0.15 + 0.95 + 2.05 -5.85 -12.25 +35.35 -31.05 = -24.45$ Step 3: Covariance: $$Cov(X,Y) = \frac{-24.45}{10} = -2.445$$ --- 5. Problem 14: Given $\sum X = 55$, $\sum Y = 74$, $\sum XY = 411$, and $n=10$, find $Cov(X,Y)$. Step 1: Means: $$\bar{X} = \frac{55}{10} = 5.5$$ $$\bar{Y} = \frac{74}{10} = 7.4$$ Step 2: Using formula: $$Cov(X,Y) = \frac{\sum XY}{n} - \bar{X}\bar{Y} = \frac{411}{10} - 5.5 \times 7.4 = 41.1 - 40.7 = 0.4$$ Final answers: - Problem 10: Positive correlation. - Problem 11: $Cov(X,Y) = 4$ - Problem 12: $Cov(X,Y) \approx 3.57$ - Problem 13: $Cov(X,Y) = -2.445$ - Problem 14: $Cov(X,Y)=0.4$