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Worked solution

Calculate correlation from six summary values

A student obtains eight pairs of sample data (x,y)(x,y) and summarises them as follows:

∑x=360,∑x2=21830,∑y=140,\sum x=360,\quad\sum x^2=21830,\quad\sum y=140, ∑y2=3500,∑xy=3985.\sum y^2=3500,\qquad\sum xy=3985.

Calculate the product moment correlation coefficient. What can you say about the relationship between xx and yy?

CJC · Discussion · Question 1

Calculate the centred sums

Answer

r=−0.952r=-0.952 (3 s.f.).

With n=8n=8,

Sxx=21830−36028=5630,S_{xx}=21830-\frac{360^2}{8}=5630,

Syy=3500−14028=1050,S_{yy}=3500-\frac{140^2}{8}=1050,

Sxy=3985−360(140)8=−2315.\begin{aligned}&S_{xy}\\ & =3985-\frac{360(140)}8\\ & =-2315.\end{aligned}

Both centred square sums are positive, so the correlation is defined:

r=−23155630(1050)=−0.9521429331….\begin{aligned}&r\\ & =\frac{-2315}{\sqrt{5630(1050)}}\\ & =-0.9521429331\ldots.\end{aligned}

This independently reproduces the printed answer −0.952-0.952. The source supplies summary values, not the eight individual pairs; no invented raw-data scatter is supplied.

How did this part go?

Describe the relationship supported by r

Answer

The sample has strong negative linear correlation.

The coefficient is close to −1, so larger x values tend to be associated with smaller y values in a strong linear pattern. This is the authored response to the source’s additional interpretation ask; the key prints only r. It neither establishes causation nor supplies enough information to inspect possible outliers or curvature without the original pairs.

How did this part go?

Solution notes

Both printed asks and all five sums with n=8 retained. Printed key gives −0.952; derivation and interpretation authored and independently checked. No source scatter or marks printed.