A company director wants to introduce a performance-related pay structure for her managers. A random sample of 15 managers is taken and the annual salary, \(y\) in \(\pounds 1000\), was recorded for each manager. The director then calculated a performance score, \(x\), for each of these managers.
The results are shown on the scatter diagram in Figure 1 on the next page.
- Describe the correlation between performance score and annual salary.
The results are also summarised in the following statistics.
$$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$
- Show that \(\mathrm { S } _ { x y } = 2006\)
- Find \(\mathrm { S } _ { y y }\)
- Find the product moment correlation coefficient between performance score and annual salary.
The director believes that there is a linear relationship between performance score and annual salary.
- State, giving a reason, whether or not these data are consistent with the director's belief.
- Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
- Give an interpretation of the value of \(b\).
- Plot your regression line on the scatter diagram in Figure 1
The director hears that one of the managers in the sample seems to be underperforming.
- On the scatter diagram, circle the point that best identifies this manager.
The director decides to use this regression line for the new performance related pay structure.
- Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-15_1390_1408_299_187}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
\includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_2654_99_115_9}
Annual salary (£1000)
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Only use this scatter diagram if you need to redraw your line.}
\includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_1378_1143_402_468}
\end{figure}