2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\).
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\caption{Fig. 2}
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- Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data.
Summary statistics for the data are as follows.
\(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\) - In this question you must show detailed reasoning.
Calculate the product moment correlation coefficient between \(x\) and \(y\).
- Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
- Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.