1. **Problem Statement:** We are given data on sunlight hours per day and corresponding plant heights. We need to calculate the correlation between sunlight and height and fit a regression line to predict plant height from sunlight duration. 2. **Formulas and Rules:** - Correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It is calculated as: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ - The regression line predicting $y$ from $x$ is: $$y = b_0 + b_1 x$$ where slope $b_1 = \frac{S_{xy}}{S_{xx}}$ and intercept $b_0 = \bar{y} - b_1 \bar{x}$. - $S_{xy} = \sum xy - \frac{\sum x \sum y}{n}$ and $S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$. 3. **Data:** Sunlight $x$: 3,4,5,6,7,3,4,6,5,7 Height $y$: 22,28,35,40,45,20,27,42,36,47 4. **Calculate sums:** $$\sum x = 3+4+5+6+7+3+4+6+5+7 = 50$$ $$\sum y = 22+28+35+40+45+20+27+42+36+47 = 342$$ $$\sum x^2 = 3^2+4^2+5^2+6^2+7^2+3^2+4^2+6^2+5^2+7^2 = 269$$ $$\sum y^2 = 22^2+28^2+35^2+40^2+45^2+20^2+27^2+42^2+36^2+47^2 = 12454$$ $$\sum xy = 3\times22 + 4\times28 + 5\times35 + 6\times40 + 7\times45 + 3\times20 + 4\times27 + 6\times42 + 5\times36 + 7\times47 = 1814$$ 5. **Calculate $S_{xy}$ and $S_{xx}$:** $$S_{xy} = 1814 - \frac{50 \times 342}{10} = 1814 - 1710 = 104$$ $$S_{xx} = 269 - \frac{50^2}{10} = 269 - 250 = 19$$ 6. **Calculate slope $b_1$ and intercept $b_0$:** $$b_1 = \frac{S_{xy}}{S_{xx}} = \frac{104}{19} \approx 5.474$$ $$\bar{x} = \frac{50}{10} = 5$$ $$\bar{y} = \frac{342}{10} = 34.2$$ $$b_0 = 34.2 - 5.474 \times 5 = 34.2 - 27.37 = 6.83$$ 7. **Regression line:** $$y = 6.83 + 5.474 x$$ 8. **Calculate correlation coefficient $r$:** $$r = \frac{10 \times 1814 - 50 \times 342}{\sqrt{(10 \times 269 - 50^2)(10 \times 12454 - 342^2)}} = \frac{18140 - 17100}{\sqrt{(2690 - 2500)(124540 - 116964)}} = \frac{1040}{\sqrt{190 \times 7576}} = \frac{1040}{\sqrt{1,438,440}} = \frac{1040}{1199.35} \approx 0.867$$ 9. **Interpretation:** The correlation coefficient $r \approx 0.867$ indicates a strong positive linear relationship between sunlight hours and plant height. The regression line can be used to predict plant height from sunlight exposure. **Final answer:** Correlation coefficient $r \approx 0.867$ Regression line: $$y = 6.83 + 5.474 x$$