Question: User: x₁\ty₁\n2\t14\n4\t28\n6\t42\n8\t56\n\nLinear Regression\ny = 7x + 0\n\nSTATISTICS\nR² = 1\nr = 1\n\nRESIDUALS\ne₁\nplot\n\nGraph: four plotted points at (2,14), (4,28), (6,42), and (8,56), with a blue linear regression line y = 7x + 0 passing through them. position_hint = center
1. **Problem Statement:** Given the data points $(2,14)$, $(4,28)$, $(6,42)$, and $(8,56)$, find the linear regression line that best fits the data.
2. **Formula for Linear Regression Line:** The equation of the regression line is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept.
3. **Calculate the slope $m$:** The slope is given by $$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$ where $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively.
4. **Calculate means:**
$$\bar{x} = \frac{2 + 4 + 6 + 8}{4} = \frac{20}{4} = 5$$
$$\bar{y} = \frac{14 + 28 + 42 + 56}{4} = \frac{140}{4} = 35$$
5. **Calculate numerator for $m$:**
$$\sum (x_i - \bar{x})(y_i - \bar{y}) = (2-5)(14-35) + (4-5)(28-35) + (6-5)(42-35) + (8-5)(56-35)$$
$$= (-3)(-21) + (-1)(-7) + (1)(7) + (3)(21) = 63 + 7 + 7 + 63 = 140$$
6. **Calculate denominator for $m$:**
$$\sum (x_i - \bar{x})^2 = (2-5)^2 + (4-5)^2 + (6-5)^2 + (8-5)^2 = 9 + 1 + 1 + 9 = 20$$
7. **Calculate slope $m$:**
$$m = \frac{140}{20} = 7$$
8. **Calculate intercept $b$:**
$$b = \bar{y} - m\bar{x} = 35 - 7 \times 5 = 35 - 35 = 0$$
9. **Final regression line:**
$$y = 7x + 0$$
10. **Interpretation:** The regression line perfectly fits the data points, as indicated by $R^2 = 1$ and correlation coefficient $r = 1$, meaning all points lie exactly on the line.
**Answer:** The linear regression line is $$y = 7x + 0$$.