1 The times, in seconds, taken by ten randomly selected competitors for the first and last sections of an Olympic bobsleigh run are denoted by \(x\) and \(y\) respectively. Summary statistics for these data are as follows.
$$\Sigma x = 113.69 \quad \Sigma y = 52.81 \quad \Sigma x ^ { 2 } = 1292.56 \quad \Sigma y ^ { 2 } = 278.91 \quad \Sigma x y = 600.41 \quad n = 10$$
- Calculate the sample product moment correlation coefficient.
- Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether there is any correlation between times taken for the first and last sections of the bobsleigh run.
- State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
- A commentator says that in order to have a fast time on the last section, you must have a fast time on the first section. Comment briefly on this suggestion.
- (A) Would your conclusion in part (ii) have been different if you had carried out the hypothesis test at the \(1 \%\) level rather than the \(10 \%\) level? Explain your answer.
(B) State one advantage and one disadvantage of using a \(1 \%\) significance level rather than a \(10 \%\) significance level in a hypothesis test.