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Difference between revisions of "1988 AHSME Problems/Problem 29"

(Created page with "==Problem== You plot weight <math>(y)</math> against height <math>(x)</math> for three of your friends and obtain the points <math>(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})</...")
 
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==Solution==
 
==Solution==
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Apply one of the standard formulae for the gradient of the line of best fit, e.g. <math>\frac{\frac{\sum {x_i y_i}}{n} - \bar{x} \bar{y}}{\frac{\sum {x_{i}^2}}{n} - \bar{x}^2}</math>, and substitute in the given condition <math>x_3 - x_2 = x_2 - x_1</math>. The answer is <math>\boxed{\text{A}}</math>.
  
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== See also ==
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{{AHSME box|year=1988|num-b=28|num-a=30}} 
  
  
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== See also ==
 
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{{AHSME box|year=1988|num-b=27|num-a=29}} 
 
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[[Category: Intermediate Statistics Problems]]
 
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:03, 27 February 2018

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

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 $\boxed{\text{A}}$.

See also

1988 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 28 		Followed by
Problem 30
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions


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