1988 AHSME Problems/Problem 29

Problem

You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$ . If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$ , which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.

$\textbf{(A)}\ \frac{y_{3}-y_{1}}{x_{3}-x_{1}}\qquad \textbf{(B)}\ \frac{(y_{2}-y_{1})-(y_{3}-y_{2})}{x_{3}-x_{1}}\qquad\\ \textbf{(C)}\ \frac{2y_{3}-y_{1}-y_{2}}{2x_{3}-x_{1}-x_{2}}\qquad \textbf{(D)}\ \frac{y_{2}-y_{1}}{x_{2}-x_{1}}+\frac{y_{3}-y_{2}}{x_{3}-x_{2}}\qquad\\ \textbf{(E)}\ \text{none of these}$

Solution 1

Let $y = mx + b$ a prediction line, and let $d = x_3 - x_2 = x_2 - x_1$ . Then $x_1 = x_2 - d$ and $x_3 = x_2 + d$ . Let $z_1 = mx_1 + b$ , $z_2 = mx_2 + b$ and $z_3 = mx_3 + b$ be the predicted values of $y_1$ , $y_2$ , and $y_3$ , respectively. Then $z_1 = m(x_2 - d) + b = z_2 - md$ and $z_3 = m(x_2 + d) + b = z_2 + md$ .

Consider all lines that pass through $(x_2, z_2)$ . Let $r_1 = z_1 - y_1$ , $r_2 = z_2 - y_2$ , and $r_3 = z_3 - y_3$ be the three residuals. Then $r_1 + r_3 = z_1 + z_3 - y_1 - y_3 = z_2 + md + z_2 - md - y_1 - y_3 = 2z_2 - y_1 - y_3$ . So $r_1 + r_3$ is the same for all these lines, as is $r_2$ . Let $s = r_1 + r_3$ .

Since the best line passing through $(x_2, z_2)$ must minimize $r_1^2 + r_2^2 + r_3^2$ , it must minimize $r_1^2 + r_3^2 = s^2 - 2r_1(s - r_1)$ , and therefore maximize $r_1(s - r_1) = -r_1^2 + sr_1$ . Since this quadratic has a negative leading coefficient, it is maximized at $r_1 = -\frac{s}{2(-1)} = \frac{s}{2}$ . In this case, $r_1 = r_3 = \frac{s}{2}$ , so $z_1 = y_1 + \frac{s}{2}$ and $z_3 = y_3 + \frac{s}{2}$ . Also, $z_3 - z_1 = y_3 - y_1$ , and the slope of the line is $\frac{z_3 - z_1}{x_3 - x_1} = \frac{y_3 - y_1}{x_3 - x_1}}$ (Error compiling LaTeX. Unknown error_msg).

Therefore, the line of best fit must have slope \boxed{(\textbf{A}) \frac{y_3 - y_1}{x_3 - x_1}} $, as any line with a different slope could be improved by replacing it with the line with the same predicted value at$ x = x_2 $and slope$ \frac{y_3 - y_1}{x_3 - x_1}}$.

-j314andrews

Solution 2

Apply one of the standard formulae for the gradient of the line of best fit, e.g. $\frac{\frac{\sum {x_i y_i}}{n} - \bar{x} \bar{y}}{\frac{\sum {x_{i}^2}}{n} - \bar{x}^2}$ , and substitute in the given condition $x_3 - x_2 = x_2 - x_1$ . The answer is $\boxed{\text{A}}$ .

1988 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 28	Followed by Problem 30
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1988 AHSME Problems/Problem 29

Contents

Problem

Solution 1

Solution 2

See also