3 A student is investigating the relationship between the length \(x \mathrm {~mm}\) and circumference \(y \mathrm {~mm}\) of plums from a large crop. The student measures the dimensions of a random sample of 10 plums from this crop. Summary statistics for these dimensions are as follows.
$$\begin{aligned}
& \sum x = 4715 \quad \sum y = 13175 \quad \sum x ^ { 2 } = 2237725
& \sum y ^ { 2 } = 17455825 \quad \sum x y = 6235575 \quad n = 10
\end{aligned}$$
- Calculate the sample product moment correlation coefficient.
- Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any correlation between length and circumference of plums from this crop. State your hypotheses clearly, defining any symbols which you use.
- (A) Explain the meaning of a 5\% significance level.
(B) State one advantage and one disadvantage of using a \(1 \%\) significance level rather than a \(5 \%\) significance level in a hypothesis test.
The student decides to take another random sample of 10 plums. Using the same hypotheses as in part (ii), the correlation coefficient for this second sample is significant at the \(5 \%\) level. The student decides to ignore the first result and concludes that there is correlation between the length and circumference of plums in the crop. - Comment on the student's decision to ignore the first result. Suggest a better way in which the student could proceed.