3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows.
$$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$
- Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
- Comment briefly on this value in this context.
- A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
- Calculate the equation of the regression line of \(y\) on \(x\).
- Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.