3. Barbara is investigating the relationship between average income (GDP per capita), \(x\) US dollars, and average annual carbon dioxide ( \(\mathrm { CO } _ { 2 }\) ) emissions, \(y\) tonnes, for different countries.
She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual \(\mathrm { CO } _ { 2 }\) emissions and average income to be 0.446
- Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero.
Barbara believes that a non-linear model would be a better fit to the data.
She codes the data using the coding \(m = \log _ { 10 } x\) and \(c = \log _ { 10 } y\) and obtains the model \(c = - 1.82 + 0.89 m\)
The product moment correlation coefficient between \(c\) and \(m\) is found to be 0.882 - Explain how this value supports Barbara's belief.
- Show that the relationship between \(y\) and \(x\) can be written in the form \(y = a x ^ { n }\) where \(a\) and \(n\) are constants to be found.