5 A student wants to know if there is a positive correlation between the amounts of two pollutants, sulphur dioxide and PM10 particulates, on different days in the area of London in which he lives; these amounts, measured in suitable units, are denoted by \(s\) and \(p\) respectively.
He uses a government website to obtain data for a random sample of 15 days on which the amounts of these pollutants were measured simultaneously. Fig. 5.1 is a scatter diagram showing the data. Summary statistics for these 15 values of \(s\) and \(p\) are as follows.
\(\sum s _ { 1 } = 155.4 \quad \sum p = 518.9 \quad \sum s ^ { 2 } = 2322.7 \quad \sum p ^ { 2 } = 21270.5 \quad \sum s p = 6009.1\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-4_935_1134_683_260}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{figure}
- Explain why the student might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
- Find the value of Pearson's product moment correlation coefficient.
- Carry out a test at the \(5 \%\) significance level to investigate whether there is positive correlation between the amounts of sulphur dioxide and PM10 particulates.
- Explain why the student made sure that the sample chosen was a random sample.
The student also wishes to model the relationship between the amounts of nitrogen dioxide \(n\) and PM10 particulates \(p\).
He takes a random sample of 54 values of the two variables, both measured at the same times. Fig. 5.2 is a scatter diagram which shows the data, together with the regression line of \(n\) on \(p\), the equation of the regression line and the value of \(r ^ { 2 }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-5_824_1230_495_258}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\end{figure} - Predict the value of \(n\) for \(p = 150\).
- Discuss the reliability of your prediction in part (e).