- A biologist is studying bears. The biologist records the length, \(d \mathrm {~cm}\), and the girth, \(g \mathrm {~cm}\), of 8 bears. The biologist summarises the data as follows
$$\begin{gathered}
\sum d = 1456.8 \quad \sum g = 713.2 \quad \sum d g = 141978.84 \quad \sum g ^ { 2 } = 72675.98 \\
S _ { d d } = 16769.78
\end{gathered}$$
- Calculate the exact value of \(S _ { d g }\) and the exact value of \(S _ { g g }\)
- Calculate the value of the product moment correlation coefficient between \(d\) and \(g\)
- Show that the equation of the regression line of \(g\) on \(d\) can be written as
$$g = - 42.3 + 0.722 d$$
where the values of the intercept and gradient are given to 3 significant figures.
- Give an interpretation, in context, of the gradient of the regression line.
Using the equation of the regression line given in part (c)
- estimate the girth of a bear with a length of 2.5 metres,
- explain why an estimate for the girth of a bear with a length of 0.5 metres is not reliable.
Using the regression line from part (c), the biologist estimates that for each \(x \mathrm {~cm}\) increase in the length of a bear there will be a 17.3 cm increase in the girth.
- Find the value of \(x\)