1. Given data for father's height $X$ and son's height $Y$: $$X = \{63, 65, 66, 67, 67, 68\}$$ $$Y = \{66, 68, 65, 67, 69, 70\}$$ Calculate means: $$\bar{X} = \frac{63 + 65 + 66 + 67 + 67 + 68}{6} = \frac{396}{6} = 66$$ $$\bar{Y} = \frac{66 + 68 + 65 + 67 + 69 + 70}{6} = \frac{405}{6} = 67.5$$ Calculate deviations and products: $$\sum (X_i - \bar{X})(Y_i - \bar{Y}) = (63-66)(66-67.5) + (65-66)(68-67.5) + (66-66)(65-67.5) + (67-66)(67-67.5) + (67-66)(69-67.5) + (68-66)(70-67.5)$$ $$= (-3)(-1.5) + (-1)(0.5) + (0)(-2.5) + (1)(-0.5) + (1)(1.5) + (2)(2.5)$$ $$= 4.5 - 0.5 + 0 - 0.5 + 1.5 + 5 = 10$$ Calculate sums of squares: $$\sum (X_i - \bar{X})^2 = (-3)^2 + (-1)^2 + 0^2 + 1^2 + 1^2 + 2^2 = 9 + 1 + 0 + 1 + 1 + 4 = 16$$ $$\sum (Y_i - \bar{Y})^2 = (-1.5)^2 + 0.5^2 + (-2.5)^2 + (-0.5)^2 + 1.5^2 + 2.5^2 = 2.25 + 0.25 + 6.25 + 0.25 + 2.25 + 6.25 = 17.5$$ Correlation coefficient: $$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} = \frac{10}{\sqrt{16 \times 17.5}} = \frac{10}{\sqrt{280}} = \frac{10}{16.7332} \approx 0.5976$$ 2. Regression line of $Y$ on $X$: $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} = \frac{10}{16} = 0.625$$ $$a = \bar{Y} - b \bar{X} = 67.5 - 0.625 \times 66 = 67.5 - 41.25 = 26.25$$ Regression equation: $$Y = a + bX = 26.25 + 0.625X$$ Predict son's height for father's height $X=70$: $$Y = 26.25 + 0.625 \times 70 = 26.25 + 43.75 = 70$$