Question 6 - A-Level Maths

AQA S1 2009 January — Question 6 15 marks

Exam Board	AQA
Module	S1 (Statistics 1)
Year	2009
Session	January
Marks	15
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear regression
Type	Calculate y on x from raw data table
Difficulty	Moderate -0.3 This is a standard S1 linear regression question requiring calculation of regression line from summary statistics (Sxx, Sxy), plotting, and residual analysis. While it involves multiple parts and arithmetic computation, all techniques are routine textbook procedures with no conceptual challenges or novel problem-solving required. Slightly easier than average due to straightforward data and clear structure.
Spec	2.02c Scatter diagrams and regression lines 5.09a Dependent/independent variables 5.09b Least squares regression: concepts 5.09c Calculate regression line 5.09e Use regression: for estimation in context

6 [Figure 1, printed on the insert, is provided for use in this question.]
For a random sample of 10 patients who underwent hip-replacement operations, records were kept of their ages, $x$ years, and of the number of days, $y$, following their operations before they were able to walk unaided safely.

Patient	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$
$\boldsymbol { x }$	55	51	62	66	72	59	78	55	62	70
$\boldsymbol { y }$	34	33	39	49	48	43	51	41	46	51

On Figure 1, complete the scatter diagram for these data.
Calculate the equation of the least squares regression line of $y$ on $x$.
Draw your regression line on Figure 1.
In fact, patients H, I and J were males and the other 7 patients were females.
1. Calculate the mean of the residuals for the 3 male patients.
2. Hence estimate, for a male patient aged 65 years, the number of days following his hip-replacement operation before he is able to walk unaided safely.

Show mark scheme Show mark scheme source

6(a)

Answer	Marks	Guidance
Figure 1: 3 correct labelled points; 2 correct labelled points	B2 (B1)	2

6(b)

Answer	Marks	Guidance
$b$ (gradient) = 0.685; $b$ (gradient) = 0.68 to 0.69; $a$ (intercept) = 0.344; $a$ (intercept) = 0.34 to 0.35	B2 (B1) (B2) (B1)
OR Attempt at $\sum x$, $\sum x^2$, $\sum y$ & $\sum xy$ or Attempt at $S_{xx}$ & $S_{xy}$; Attempt at correct formula for $b$ (gradient) $b$ (gradient) = 0.685; $a$ (intercept) = 0.344$; Accept $a$ & $b$ interchanged only if then identified correctly by a stated or used equation in (c) or (d)	M1 (m1) (A1) (A1)
4

6(c)

Answer	Marks	Guidance
Figure 1: Correct line $(50, 34$ to $35)$ $(60, 40\frac{1}{2}$ to $42)$ $(70, 47\frac{3}{4}$ to $49)$ $(80, 54$ to $56)$; If B0 but evidence of use of line for ≥ 2 points within range $50 \leq x \leq 80$	B2dep (M1)	2

6(d)(i)

Answer	Marks	Guidance
Residual = $y - (a + bx)$ [or $(a + bx) - y$] $H$: $2.5$ to $4.0(0)$; $I$: $2.5$ to $4.0(0)$; $J$: $2.0(0)$ to $4.0(0)$; Mean = $2.3$ to $4.0(0)$	M1 (A2,1) (–1 EE) (A1dep)	4

6(d)(ii)

Answer	Marks	Guidance
$y_{65} = a + b \times 65$ or $y_{65} = 44$ to $45.5$ $+ [(d)(i)]$ or $[2.95$ to $2.97]$ $= 46$ to $50$	M1 (m1) (A1)	3
Special Cases: Line drawn/calc' on H, I & J or linear interp' using I & J = 47 to 49; 44 to 45.5 seen with no evidence ⟹ B1	B2

Total for Q6: 15 marks

**6(a)**

| Figure 1: 3 correct labelled points; 2 correct labelled points | B2 (B1) | 2 | Deduct 1 mark if not labelled |

**6(b)**

| $b$ (gradient) = 0.685; $b$ (gradient) = 0.68 to 0.69; $a$ (intercept) = 0.344; $a$ (intercept) = 0.34 to 0.35 | B2 (B1) (B2) (B1) | | AWRT (0.68502); AWFW; AWRT (0.34404); AWFW |
| OR Attempt at $\sum x$, $\sum x^2$, $\sum y$ & $\sum xy$ or Attempt at $S_{xx}$ & $S_{xy}$; Attempt at correct formula for $b$ (gradient) $b$ (gradient) = 0.685; $a$ (intercept) = 0.344$; Accept $a$ & $b$ interchanged only if then identified correctly by a stated or used equation in (c) or (d) | M1 (m1) (A1) (A1) | | 630, 40344, 435 & 27853 (all 4 attempted) or 654 & 448 (both attempted); AWRT; AWRT |
| | | 4 | |

**6(c)**

| Figure 1: Correct line $(50, 34$ to $35)$ $(60, 40\frac{1}{2}$ to $42)$ $(70, 47\frac{3}{4}$ to $49)$ $(80, 54$ to $56)$; If B0 but evidence of use of line for ≥ 2 points within range $50 \leq x \leq 80$ | B2dep (M1) | 2 | Dep on ≥ B1 B1 or ≥A1 A0 in (b); At least from x = 55 to 70; Any two; Calc' or points shown on graph |

**6(d)(i)**

| Residual = $y - (a + bx)$ [or $(a + bx) - y$] $H$: $2.5$ to $4.0(0)$; $I$: $2.5$ to $4.0(0)$; $J$: $2.0(0)$ to $4.0(0)$; Mean = $2.3$ to $4.0(0)$ | M1 (A2,1) (–1 EE) (A1dep) | 4 | Used or implied; or equivalent; (using graph): ≥ 1 residual correct (2.98); AWFW; ignore signs only (3.19); providing all the same (2.70); AWFW; do not ignore sign (2.96); Dep on previous A2 scored |

**6(d)(ii)**

| $y_{65} = a + b \times 65$ or $y_{65} = 44$ to $45.5$ $+ [(d)(i)]$ or $[2.95$ to $2.97]$ $= 46$ to $50$ | M1 (m1) (A1) | 3 | Use shown or AWFW (44.9); Use shown or AWFW; ignore sign of mean residual (47.8); AWFW |
| Special Cases: Line drawn/calc' on H, I & J or linear interp' using I & J = 47 to 49; 44 to 45.5 seen with no evidence ⟹ B1 | B2 | | $y_{65} = 4.51 + 0.666x ⟹ 47.8$ or no evidence of method [from (d)(i)] and/or (d)(ii)]; Evidence of incorrect method ⟹ B0 |

**Total for Q6: 15 marks**

---

Show LaTeX source

6 [Figure 1, printed on the insert, is provided for use in this question.]\\
For a random sample of 10 patients who underwent hip-replacement operations, records were kept of their ages, $x$ years, and of the number of days, $y$, following their operations before they were able to walk unaided safely.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Patient & $\mathbf { A }$ & $\mathbf { B }$ & $\mathbf { C }$ & $\mathbf { D }$ & $\mathbf { E }$ & $\mathbf { F }$ & $\mathbf { G }$ & $\mathbf { H }$ & $\mathbf { I }$ & $\mathbf { J }$ \\
\hline
$\boldsymbol { x }$ & 55 & 51 & 62 & 66 & 72 & 59 & 78 & 55 & 62 & 70 \\
\hline
$\boldsymbol { y }$ & 34 & 33 & 39 & 49 & 48 & 43 & 51 & 41 & 46 & 51 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, complete the scatter diagram for these data.
\item Calculate the equation of the least squares regression line of $y$ on $x$.
\item Draw your regression line on Figure 1.
\item In fact, patients H, I and J were males and the other 7 patients were females.
\begin{enumerate}[label=(\roman*)]
\item Calculate the mean of the residuals for the 3 male patients.
\item Hence estimate, for a male patient aged 65 years, the number of days following his hip-replacement operation before he is able to walk unaided safely.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2009 Q6 [15]}}

This paper (7 questions)

View full paper

Q1 7 Q2 7 Q3 14 Q4 12 Q5 8 Q6 15 Q7 12

Patient	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)	\(\mathbf { G }\)	\(\mathbf { H }\)	\(\mathbf { I }\)	\(\mathbf { J }\)
\(\boldsymbol { x }\)	55	51	62	66	72	59	78	55	62	70
\(\boldsymbol { y }\)	34	33	39	49	48	43	51	41	46	51

Answer	Marks	Guidance
\(b\) (gradient) = 0.685; \(b\) (gradient) = 0.68 to 0.69; \(a\) (intercept) = 0.344; \(a\) (intercept) = 0.34 to 0.35	B2 (B1) (B2) (B1)
OR Attempt at \(\sum x\), \(\sum x^2\), \(\sum y\) & \(\sum xy\) or Attempt at \(S_{xx}\) & \(S_{xy}\); Attempt at correct formula for \(b\) (gradient) \(b\) (gradient) = 0.685; \(a\) (intercept) = 0.344\(; Accept \)a\( & \)b$ interchanged only if then identified correctly by a stated or used equation in (c) or (d)	M1 (m1) (A1) (A1)
4

Answer	Marks	Guidance
\(y_{65} = a + b \times 65\) or \(y_{65} = 44\) to \(45.5\) \(+ [(d)(i)]\) or \([2.95\) to \(2.97]\) \(= 46\) to \(50\)	M1 (m1) (A1)	3
Special Cases: Line drawn/calc' on H, I & J or linear interp' using I & J = 47 to 49; 44 to 45.5 seen with no evidence ⟹ B1	B2