1. **State the problem:** We are given sums of $x$, $x^2$, $y$, $y^2$, and $xy$ for seven data points and need to find the linear regression line $\bar{y} = a + bx$. 2. **Recall the formulas:** The slope $b$ and intercept $a$ of the regression line are given by: $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$a = \frac{\sum y}{n} - b \frac{\sum x}{n}$$ where $n$ is the number of data points. 3. **Identify given values:** $$n = 7$$ $$\sum x = 117$$ $$\sum x^2 = 1313$$ $$\sum y = 260$$ $$\sum y^2 = 6580$$ $$\sum xy = 2827$$ 4. **Calculate slope $b$:** $$b = \frac{7 \times 2827 - 117 \times 260}{7 \times 1313 - 117^2} = \frac{19789 - 30420}{9191 - 13689} = \frac{-11631}{-4498} = 2.586$$ 5. **Calculate intercept $a$:** $$a = \frac{260}{7} - 2.586 \times \frac{117}{7} = 37.143 - 2.586 \times 16.714 = 37.143 - 43.214 = -6.071$$ 6. **Write the regression equation:** $$\bar{y} = -6.071 + 2.586x$$ This means for each unit increase in $x$, $y$ increases by approximately 2.586, starting from about -6.071 when $x=0$.