5 Two practice GCSE examinations in mathematics are given to all of the students in a large year group. A teacher wants to check whether there is a positive relationship between the marks obtained by the students in the two examinations. She selects a random sample of 20 students. Summary data for the marks obtained in the first and second practice examinations, \(x\) and \(y\) respectively, are as follows. $$\sum x = 565 \quad \sum y = 724 \quad \sum x ^ { 2 } = 17103 \quad \sum y ^ { 2 } = 29286 \quad \sum x y = 21635$$ The teacher decides to carry out a hypothesis test based on Pearson’s product moment correlation coefficient.

1. In this question you must show detailed reasoning. Calculate the value of Pearson's product moment correlation coefficient.
2. Carry out the test at the \(5 \%\) significance level.
3. Given that the teacher did not draw a scatter diagram before carrying out the test, comment on the validity of the test.