2. A large company is analysing how much money it spends on paper in its offices each year. The number of employees in the office, \(x\), and the amount spent on paper in a year, \(p\) (\$ hundreds), in each of 12 randomly selected offices were recorded.
The results are summarised in the following statistics.
$$\sum x = 93 \quad \mathrm {~S} _ { x x } = 148.25 \quad \sum p = 273 \quad \sum p ^ { 2 } = 6602.72 \quad \sum x p = 2347$$
- Show that \(\mathrm { S } _ { x p } = 231.25\)
- Find the product moment correlation coefficient for these data.
- Find the equation of the regression line of \(p\) on \(x\) in the form \(p = a + b x\)
- Give an interpretation of the gradient of your regression line.
The director of the company wants to reduce the amount spent on paper each year.
He wants each office to aim for a model of the form \(p = \frac { 4 } { 5 } a + \frac { 1 } { 2 } b x\), where \(a\) and \(b\) are the values found in part (c).
Using the data for the 93 employees from the 12 offices,
- estimate the percentage saving in the amount spent on paper each year by the company using the director's model.