A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.
| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \(x\) | 1.2 | 1.4 | 0.9 | 1.1 | 0.8 | 1.0 | 0.6 | 1.5 |
| \(y\) | 0.3 | 0.4 | 0.6 | 0.6 | 0.25 | 0.75 | 0.6 | 0.35 |
- Calculate the product moment correlation coefficient for this sample.
- Stating your hypotheses, test, at the \(2.5 \%\) significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question.
Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be - 0.431 . A test, at the \(5 \%\) significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.
- Find the range of possible values of \(N\).
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