Edexcel S1 2021 January — Question 5 17 marks

Exam BoardEdexcel
ModuleS1 (Statistics 1)
Year2021
SessionJanuary
Marks17
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear regression
TypeInterpret correlation strength/direction
DifficultyModerate -0.8 This is a standard S1 linear regression question requiring routine application of formulas (Sxy, Syy, PMCC, regression line) with straightforward interpretation tasks. All parts follow textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step calculation requirements.
Spec2.02c Scatter diagrams and regression lines5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09c Calculate regression line5.09e Use regression: for estimation in context
  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}
Question 5:
Part (a):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Positive (correlation) or e.g. "salary (\(y\)) increases as performance (\(x\)) increases"B1 "Positive skew" is B0
Part (b)(i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(19428 - \frac{465\times562}{15}\) or \(19428 - \frac{261330}{15} = 2006\)B1cso All correct values must be seen
Part (b)(ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\left[S_{yy}=\right]\ 23140 - \frac{562^2}{15}\)M1 Correct expression
\(= 2083.7333...\) awrt 2080A1 Allow \(\frac{31256}{15}\); expect to see 2084
Part (c):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\left[r=\right] \frac{2006}{\sqrt{2492\times\text{"2083.73..."}}}\); \(= 0.8803104...\) awrt 0.880M1;A1 M1 for correct expression but ft their \(S_{yy}\neq23140\), or answer only of 0.88
Part (d):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Is consistent and the points on the scatter diagram lie close to a straight line, or \(r\) is close to 1, or strong/high (positive) correlationB1 [no ft] for "yes" (o.e.) and a suitable reason based on scatter diagram or value of \(r\)
Part (e):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(b = \frac{2006}{2492} = 0.80[497...]\); \(a = 37.46...-\text{"b"}\times31\ [=12.512...]\)M1;A1;M1 1st A1 for \(b=0.80\) or better (allow \(\frac{1003}{1246}\) but not \(\frac{2006}{2492}\)). 2nd M1 for correct expression for \(a\)
\(y = 12.5 + 0.805x\)A1 2nd A1 for correct equation with \(b=\) awrt 0.805 and \(a=\) awrt 12.5
Part (f):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
An increase of 1 (performance) point gives an extra £800 (1 sf) in salaryB1 Comment mentioning value in £ of \(b\times1000\) (awrt 1 sf) per performance point. Condone use of \$ rather than £
Part (g):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Line must cross \(x=9\) and \(x=50\) to score either mark
Line for 9–50: Intercept at "12.5" (accept 11.5–13.5)B1ft 1st B1ft for correct intercept for their line (\(\pm1\))
Line for 9–50: At \(x=50\), \(y=52.8\) (accept 52–54)B1 2nd B1 for \(y=52\sim54\) when \(x=50\)
Part (h):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
For the point (25, 48) circledB1 If more than one of the given points circled, B0
Part (i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\text{"12.5"} + 30\times\text{"0.805"}\ [= 36\sim37]\) or allow 2sf from their diagramM1 M1 for using \(x=30\) in their equation ft their \(a\) and \(b\) to any accuracy
Salary of awrt £36 700 (or 36.7 thousands)A1 Answer only of awrt 37 000 can score M1A0 
# Question 5:

## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Positive (correlation) or e.g. "salary ($y$) increases as performance ($x$) increases" | B1 | "Positive skew" is B0 |

## Part (b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $19428 - \frac{465\times562}{15}$ or $19428 - \frac{261330}{15} = 2006$ | B1cso | All correct values must be seen |

## Part (b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[S_{yy}=\right]\ 23140 - \frac{562^2}{15}$ | M1 | Correct expression |
| $= 2083.7333...$ awrt **2080** | A1 | Allow $\frac{31256}{15}$; expect to see 2084 |

## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[r=\right] \frac{2006}{\sqrt{2492\times\text{"2083.73..."}}}$; $= 0.8803104...$ awrt **0.880** | M1;A1 | M1 for correct expression but ft their $S_{yy}\neq23140$, or answer only of 0.88 |

## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Is consistent **and** the points on the scatter diagram lie close to a straight line, or $r$ is close to 1, or strong/high (positive) correlation | B1 | [no ft] for "yes" (o.e.) **and** a suitable reason based on scatter diagram or value of $r$ |

## Part (e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b = \frac{2006}{2492} = 0.80[497...]$; $a = 37.46...-\text{"b"}\times31\ [=12.512...]$ | M1;A1;M1 | 1st A1 for $b=0.80$ or better (allow $\frac{1003}{1246}$ but not $\frac{2006}{2492}$). 2nd M1 for correct expression for $a$ |
| $y = 12.5 + 0.805x$ | A1 | 2nd A1 for correct equation with $b=$ awrt 0.805 and $a=$ awrt 12.5 |

## Part (f):
| Answer/Working | Marks | Guidance |
|---|---|---|
| An increase of 1 (performance) point gives an extra £800 (1 sf) in salary | B1 | Comment mentioning value in £ of $b\times1000$ (awrt 1 sf) per performance point. Condone use of \$ rather than £ |

## Part (g):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Line must cross $x=9$ and $x=50$ to score either mark | | |
| Line for 9–50: Intercept at "12.5" (accept 11.5–13.5) | B1ft | 1st B1ft for correct intercept for their line ($\pm1$) |
| Line for 9–50: At $x=50$, $y=52.8$ (accept 52–54) | B1 | 2nd B1 for $y=52\sim54$ when $x=50$ |

## Part (h):
| Answer/Working | Marks | Guidance |
|---|---|---|
| For the point (25, 48) circled | B1 | If more than one of the given points circled, B0 |

## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{"12.5"} + 30\times\text{"0.805"}\ [= 36\sim37]$ or allow 2sf from their diagram | M1 | M1 for using $x=30$ in their equation ft their $a$ and $b$ to any accuracy |
| Salary of awrt £**36 700** (or 36.7 thousands) | A1 | Answer only of awrt 37 000 can score M1A0 |

---
\begin{enumerate}
  \item 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.\\
(a) Describe the correlation between performance score and annual salary.
\end{enumerate}

The results are also summarised in the following statistics.

$$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$

(b) (i) Show that $\mathrm { S } _ { x y } = 2006$\\
(ii) Find $\mathrm { S } _ { y y }$\\
(c) 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.\\
(d) State, giving a reason, whether or not these data are consistent with the director's belief.\\
(e) 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.\\
(f) Give an interpretation of the value of $b$.\\
(g) 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.\\
(h) 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.\\
(i) Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-15_1390_1408_299_187}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

\includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_2654_99_115_9}

Annual salary (£1000)

\begin{figure}[h]
\begin{center}
\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{center}
\end{figure}

\hfill \mbox{\textit{Edexcel S1 2021 Q5 [17]}}
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