1. **Problem Statement:** Fit the line of regression of $y$ on $x$ for the given data: $x = [0,1,2,3,4,5,6,7,8,9]$ $y = [43,46,82,98,100,125,150,200,210,250]$ Then calculate the predicted number of bacterial cells after 15 hours. 2. **Formula and Important Rules:** The line of regression of $y$ on $x$ is given by: $$y = a + bx$$ where $$b = \frac{S_{xy}}{S_{xx}} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ and $$a = \bar{y} - b\bar{x}$$ Here, $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively. 3. **Calculate Means:** $$\bar{x} = \frac{0+1+2+3+4+5+6+7+8+9}{10} = \frac{45}{10} = 4.5$$ $$\bar{y} = \frac{43+46+82+98+100+125+150+200+210+250}{10} = \frac{1304}{10} = 130.4$$ 4. **Calculate $S_{xx}$ and $S_{xy}$:** Calculate each $(x_i - \bar{x})$ and $(y_i - \bar{y})$: | $x_i$ | $y_i$ | $x_i - \bar{x}$ | $y_i - \bar{y}$ | $(x_i - \bar{x})^2$ | $(x_i - \bar{x})(y_i - \bar{y})$ | |-------|-------|-----------------|-----------------|---------------------|-----------------------------| | 0 | 43 | 0 - 4.5 = -4.5 | 43 - 130.4 = -87.4 | 20.25 | (-4.5)(-87.4) = 393.3 | | 1 | 46 | -3.5 | -84.4 | 12.25 | 295.4 | | 2 | 82 | -2.5 | -48.4 | 6.25 | 121.0 | | 3 | 98 | -1.5 | -32.4 | 2.25 | 48.6 | | 4 | 100 | -0.5 | -30.4 | 0.25 | 15.2 | | 5 | 125 | 0.5 | -5.4 | 0.25 | -2.7 | | 6 | 150 | 1.5 | 19.6 | 2.25 | 29.4 | | 7 | 200 | 2.5 | 69.6 | 6.25 | 174.0 | | 8 | 210 | 3.5 | 79.6 | 12.25 | 278.6 | | 9 | 250 | 4.5 | 119.6 | 20.25 | 538.2 | Sum these values: $$S_{xx} = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 82.5$$ $$S_{xy} = 393.3 + 295.4 + 121.0 + 48.6 + 15.2 - 2.7 + 29.4 + 174.0 + 278.6 + 538.2 = 1890.0$$ 5. **Calculate slope $b$ and intercept $a$:** $$b = \frac{S_{xy}}{S_{xx}} = \frac{1890.0}{82.5} = 22.9091$$ $$a = \bar{y} - b\bar{x} = 130.4 - 22.9091 \times 4.5 = 130.4 - 103.0909 = 27.3091$$ 6. **Regression line equation:** $$y = 27.3091 + 22.9091x$$ 7. **Predict number of bacterial cells after 15 hours:** Substitute $x=15$: $$y = 27.3091 + 22.9091 \times 15 = 27.3091 + 343.6365 = 370.9456$$ So, the predicted number of bacterial cells after 15 hours is approximately 371.