The table shows the amount, \(x\), in hundreds of pounds, spent on heating and the number of absences, \(y\), at a factory during each month in 2014.
Amount, \(x\), spent on
heating (£ hundreds)
21
23
19
15
14
5
2
10
9
20
18
23
Number of absences, \(y\)
23
25
18
18
12
10
4
9
11
15
20
26
\(n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343\)
(a) Calculate \(r\), the product moment correlation coefficient, showing that \(r > 0.92\).
(b) A manager says, 'The value of \(r\) shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
The months in 2014 were numbered \(1,2,3 , \ldots , 12\). The output, \(z\), in suitable units was recorded along with the month number, \(n\), for each month in 2014. The equation of the regression line of \(z\) on \(n\) was found to be \(z = 0.6 n + 17\).
(a) Use this equation to explain whether output generally increased or decreased over these months.
(b) Find the mean of \(n\) and use the equation of the regression line to calculate the mean of \(z\).
(c) Hence calculate the total output in 2014.