1. **State the problem:** We are given data about house sizes and selling prices, and we want to find the linear regression line that predicts price based on size. The regression line is given by the equation $$y = a + bx$$ where $y$ is the selling price (in thousands), $x$ is the house size (in square feet), $b$ is the gradient (slope), and $a$ is the intercept. 2. **Given data:** - Mean house size $\bar{x} = 2200$ sq ft - Standard deviation of house size $s_x = 400$ sq ft - Mean selling price $\bar{y} = 350$ (thousands) - Standard deviation of selling price $s_y = 75$ (thousands) - Correlation coefficient $r = 0.85$ 3. **Formula for gradient $b$:** $$b = r \times \frac{s_y}{s_x}$$ 4. **Calculate $b$:** $$b = 0.85 \times \frac{75}{400} = 0.85 \times 0.1875 = 0.159375$$ Rounded to 3 decimal places: $$b = 0.159$$ 5. **Formula for intercept $a$:** $$a = \bar{y} - b \times \bar{x}$$ 6. **Calculate $a$:** $$a = 350 - 0.159 \times 2200 = 350 - 349.8 = 0.2$$ 7. **Estimate price for a 2500 sq ft house:** Use the regression equation: $$y = a + bx = 0.2 + 0.159 \times 2500 = 0.2 + 397.5 = 397.7$$ This is in thousands, so the estimated price is 397.7 (thousands). **Final answers:** - Gradient $b = 0.159$ - Intercept $a = 0.2$ - Estimated price for 2500 sq ft house = 397.7 (thousands)