Edexcel S1 2021 January — Question 5

Exam BoardEdexcel
ModuleS1 (Statistics 1)
Year2021
SessionJanuary
TopicLinear regression
TypeInterpret correlation strength/direction
  1. 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.
    1. Describe the correlation between performance score and annual salary.
    The results are also summarised in the following statistics. $$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$
    1. Show that \(\mathrm { S } _ { x y } = 2006\)
    2. Find \(\mathrm { S } _ { y y }\)
  3. 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.
  4. State, giving a reason, whether or not these data are consistent with the director's belief.
  5. 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.
  6. Give an interpretation of the value of \(b\).
  7. 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.
  8. 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.
    1. 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}
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