1. **Problem Statement:** Suppose we have data on the number of hours studied ($x$) and the corresponding exam scores ($y$) for 5 students: $$\begin{array}{c|c} \text{Hours Studied }(x) & \text{Exam Score }(y) \\ \hline 1 & 50 \\ 2 & 55 \\ 3 & 65 \\ 4 & 70 \\ 5 & 75 \end{array}$$ We want to find the linear regression line $y = mx + b$ that best fits this data. 2. **Formula and Important Rules:** The formulas for the slope $m$ and intercept $b$ of the regression line are: $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ where $n$ is the number of data points. 3. **Calculate the necessary sums:** $$n = 5$$ $$\sum x = 1 + 2 + 3 + 4 + 5 = 15$$ $$\sum y = 50 + 55 + 65 + 70 + 75 = 315$$ $$\sum xy = (1)(50) + (2)(55) + (3)(65) + (4)(70) + (5)(75) = 50 + 110 + 195 + 280 + 375 = 1010$$ $$\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$$ 4. **Calculate slope $m$:** $$m = \frac{5 \times 1010 - 15 \times 315}{5 \times 55 - 15^2} = \frac{5050 - 4725}{275 - 225} = \frac{325}{50} = 6.5$$ 5. **Calculate intercept $b$:** $$b = \frac{315 - 6.5 \times 15}{5} = \frac{315 - 97.5}{5} = \frac{217.5}{5} = 43.5$$ 6. **Final regression equation:** $$y = 6.5x + 43.5$$ This means for each additional hour studied, the exam score increases by 6.5 points on average. 7. **Interpretation:** The regression line can be used to predict exam scores based on hours studied. For example, if a student studies 3 hours, predicted score is: $$y = 6.5 \times 3 + 43.5 = 19.5 + 43.5 = 63$$ This is a simple linear regression example showing how to find the best fit line for data.