1. **Problem:** Find the regression lines of y on x and x on y for the data points given. 2. **Formula:** The regression line of y on x is given by $$y = mx + c$$ where $$m = \frac{S_{xy}}{S_{xx}}$$ and $$c = \bar{y} - m\bar{x}$$. The regression line of x on y is $$x = m'y + c'$$ where $$m' = \frac{S_{xy}}{S_{yy}}$$ and $$c' = \bar{x} - m'\bar{y}$$. 3. **Calculate means and sums for problem 1:** $$\bar{x} = \frac{1+2+3+4+5}{5} = 3$$ $$\bar{y} = \frac{3+5+7+9+11}{5} = 7$$ Calculate sums: $$S_{xx} = \sum (x_i - \bar{x})^2 = (1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2 = 4 + 1 + 0 + 1 + 4 = 10$$ $$S_{yy} = \sum (y_i - \bar{y})^2 = (3-7)^2 + (5-7)^2 + (7-7)^2 + (9-7)^2 + (11-7)^2 = 16 + 4 + 0 + 4 + 16 = 40$$ $$S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) = (1-3)(3-7) + (2-3)(5-7) + (3-3)(7-7) + (4-3)(9-7) + (5-3)(11-7) = 8 + 2 + 0 + 2 + 8 = 20$$ 4. **Regression line of y on x:** $$m = \frac{S_{xy}}{S_{xx}} = \frac{20}{10} = 2$$ $$c = \bar{y} - m\bar{x} = 7 - 2 \times 3 = 1$$ So, $$y = 2x + 1$$. 5. **Regression line of x on y:** $$m' = \frac{S_{xy}}{S_{yy}} = \frac{20}{40} = 0.5$$ $$c' = \bar{x} - m'\bar{y} = 3 - 0.5 \times 7 = -0.5$$ So, $$x = 0.5y - 0.5$$. --- Repeat similar steps for each problem: **Problem 2:** $$\bar{x} = 6, \bar{y} = 13$$ $$S_{xx} = 40, S_{yy} = 80, S_{xy} = 56$$ $$m = \frac{56}{40} = 1.4$$ $$c = 13 - 1.4 \times 6 = 4.6$$ Regression line: $$y = 1.4x + 4.6$$ --- **Problem 3:** (a) Plot points (not shown here). (b) Regression line slope and intercept: $$\bar{x} = 3.2, \bar{y} = 11.6$$ $$S_{xx} = 18.8, S_{xy} = 34.4$$ $$m = \frac{34.4}{18.8} \approx 1.83$$ $$c = 11.6 - 1.83 \times 3.2 \approx 5.83$$ Regression line: $$y = 1.83x + 5.83$$ Estimate $$y$$ at $$x=4$$: $$y = 1.83 \times 4 + 5.83 = 13.15$$ --- **Problem 4:** $$\bar{x} = 30, \bar{y} = 58$$ $$S_{xx} = 1000, S_{xy} = 850$$ $$m = \frac{850}{1000} = 0.85$$ $$c = 58 - 0.85 \times 30 = 32.5$$ Regression line: $$y = 0.85x + 32.5$$ Predict $$y$$ at $$x=35$$: $$y = 0.85 \times 35 + 32.5 = 62.25$$ --- **Problem 5:** $$\bar{x} = 4.2, \bar{y} = 11.6$$ $$S_{xx} = 5.2, S_{xy} = 11.6$$ $$m = \frac{11.6}{5.2} \approx 2.23$$ $$c = 11.6 - 2.23 \times 4.2 \approx 2.18$$ Regression line: $$y = 2.23x + 2.18$$ Predict $$y$$ at $$x=7$$: $$y = 2.23 \times 7 + 2.18 = 17.79$$ --- **Problem 6:** $$\bar{x} = 4, \bar{y} = 13$$ $$S_{xx} = 18, S_{xy} = 38$$ $$m = \frac{38}{18} \approx 2.11$$ $$c = 13 - 2.11 \times 4 = 4.56$$ Regression line: $$y = 2.11x + 4.56$$ --- **Problem 7:** $$\bar{x} = 12.5, \bar{y} = 28$$ $$S_{xx} = 125, S_{xy} = 210$$ $$m = \frac{210}{125} = 1.68$$ $$c = 28 - 1.68 \times 12.5 = 6$$ Regression line: $$y = 1.68x + 6$$ Estimate $$x$$ when $$y=30$$: $$30 = 1.68x + 6 \Rightarrow x = \frac{30-6}{1.68} = 14.29$$ --- **Problem 8:** $$\bar{x} = 4, \bar{y} = 10.2$$ $$S_{xx} = 20, S_{yy} = 74.8, S_{xy} = 38$$ $$b = \frac{S_{xy}}{S_{xx}} = 1.9$$ $$a = \bar{y} - b\bar{x} = 10.2 - 1.9 \times 4 = 2.6$$ Regression line: $$y = 2.6 + 1.9x$$ For $$x = a + by$$ regression: $$b' = \frac{S_{xy}}{S_{yy}} = 0.51$$ $$a' = \bar{x} - b'\bar{y} = 4 - 0.51 \times 10.2 = -1.2$$ Regression line: $$x = -1.2 + 0.51y$$ --- **Problem 9:** $$\bar{x} = 18, \bar{y} = 58$$ $$S_{xx} = 60, S_{xy} = 90$$ $$m = \frac{90}{60} = 1.5$$ $$c = 58 - 1.5 \times 18 = 31$$ Regression line: $$y = 1.5x + 31$$ Predict $$y$$ at $$x=20$$: $$y = 1.5 \times 20 + 31 = 61$$ --- **Problem 10:** $$\bar{x} = 5, \bar{y} = 31.6$$ $$S_{yy} = 130.8, S_{xy} = 65$$ $$m' = \frac{S_{xy}}{S_{yy}} = 0.497$$ $$c' = \bar{x} - m'\bar{y} = 5 - 0.497 \times 31.6 = -10.7$$ Regression line: $$x = 0.497y - 10.7$$ Predict $$x$$ when $$y=35$$: $$x = 0.497 \times 35 - 10.7 = 6.0$$