3 A researcher is investigating the relationship between temperature and levels of the air pollutant nitrous oxide at a particular site. The researcher believes that there will be a positive correlation between the daily maximum temperature, \(x\), and nitrous oxide level, \(y\). Data are collected for 10 randomly selected days. The data, measured in suitable units, are given in the table and illustrated on the scatter diagram.
| \(x\) | 13.3 | 17.2 | 16.9 | 18.7 | 18.4 | 19.3 | 23.1 | 15.0 | 20.6 | 14.4 |
| \(y\) | 9 | 11 | 14 | 26 | 43 | 25 | 52 | 15 | 10 | 7 |
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- Calculate the value of Spearman's rank correlation coefficient for these data.
- Perform a hypothesis test at the \(5 \%\) level to check the researcher's belief, stating your hypotheses clearly.
- It is suggested that it would be preferable to carry out a test based on the product moment correlation coefficient. State the distributional assumption required for such a test to be valid. Explain how a scatter diagram can be used to check whether the distributional assumption is likely to be valid and comment on the validity in this case.
- A statistician investigates data over a much longer period and finds that the assumptions for the use of the product moment correlation coefficient are in fact valid. Give the critical region for the test at the \(1 \%\) level, based on a sample of 60 days.
- In a different research project, into the correlation between daily temperature and ozone pollution levels, a positive correlation is found. It is argued that this shows that high temperatures cause increased ozone levels. Comment on this claim.