6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. The table shows the values of the efficiency rating, y , after the machine has carried out its task various numbers of times, \(x\)
| x | 0 | 1 | 2 | 3 | 4 | 7 | 13 | 30 |
| y | 0 | 4 | 8 | 10 | 11 | 12 | 13 | 14 |
$$\left[ n = 8 , \Sigma x = 60 , \Sigma y = 72 , \Sigma x ^ { 2 } = 1148 , \Sigma y ^ { 2 } = 810 , \Sigma x y = 767 . \right]$$
These data are illustrated in the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-4_769_1328_760_411}
- (a) Calculate the value of r , the product moment correlation coefficient.
(b) Without calculation, state with a reason the value of \(\mathrm { r } _ { \mathrm { s } ^ { \prime } }\) Spearman's rank correlation coefficient. - A researcher suggests that the data for \(\mathrm { x } = 0\) and \(\mathrm { x } = 1\) should be ignored. Without cal culation, state with a reason what effect this would have on the value of
(a) \(r\),
(b) \(r _ { s }\). - Use the diagram to estimate the value of y when \(\mathrm { x } = 29\).
- Jack finds the equation of the regression line of y on xf for all the data, and uses it to estimate the value of \(y\) when \(x = 29\). Without calculation, state with a reason whether this estimate or the one found in part (iii) will be the more reliable.