N2003 P2 Q11

A random sample of eight pairs of values of \(x\) and \(y\) is used to obtain the following equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) respectively.

\(y=-\frac{7}{10}x+\frac{151}{10}\), \(x=-\frac{7}{6}y+20\)

Seven of the pairs of data are given in the table.

Find the eighth pair of values of \(x\) and \(y\). [5]

Determine the value of the product moment correlation coefficient, and say what it leads you to expect about the scatter diagram for this sample. [4]

Let \(Y\) be the value obtained by substituting a sample value of \(x\) into the equation of the regression line of \(y\) on \(x\). Evaluate \(Y\) for each of the eight values of \(x\) and verify that \({\sum (y - Y)^2 = 8.8}\).[3]

For each of the eight sample values of \(x\), \({Y}'\) is given by \({Y}'=a+bx\), where \(a\) and \(b\) are any constants. What can you say about the value of \(\sum{{{\left( y-{Y}' \right)}^{2}}}\)?[2]