MediumMathematicsHigh School (11–12)Statistics and ProbabilityNumerical

A researcher records the age x of eight cars, in years, and the repair cost y of each in hundreds…

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A researcher records the age $x$ of eight cars, in years, and the repair cost $y$ of each in hundreds of rupees. She does not keep the individual values, only the totals: $n = 8$ , $\sum x = 76$ , $\sum y = 129$ , $\sum xy = 1324$ , $\sum x^2 = 820$ and $\sum y^2 = 2187$ . Find the three quantities $n\sum xy - \sum x\sum y$ , $n\sum x^2 - \left(\sum x\right)^2$ and $n\sum y^2 - \left(\sum y\right)^2$ , and then find $r$ .

Need a reminder? The key ideaShow

$r = \dfrac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}$ and $-1 \le r \le 1$ . From totals, the numerator is $n\sum xy - \sum x \sum y$ . The denominator has the factors $n\sum x^2 - \left(\sum x\right)^2$ and $n\sum y^2 - \left(\sum y\right)^2$ under one root. $r$ is unchanged by a change of units, and for ranks $r_s = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}$ .

The worked solution

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