4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows.
$$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$
- Calculate the value of Pearson's product-moment correlation coefficient \(r\).
- State an assumption needed to be able to carry out a significance test on the value of \(r\).
- Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
- There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by
$$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$
Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.