Question 5 - A-Level Maths

AQA S1 2014 June — Question 5 13 marks

Exam Board	AQA
Module	S1 (Statistics 1)
Year	2014
Session	June
Marks	13
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Bivariate data
Type	Calculate r from raw bivariate data
Difficulty	Moderate -0.5 This is a standard S1 correlation calculation requiring the formula r = Sxy/√(SxxSyy) with straightforward arithmetic from given data. While tedious with 12 data points, it's a routine textbook exercise testing only recall and careful calculation, making it slightly easier than average A-level difficulty.
Spec	2.02c Scatter diagrams and regression lines 5.08a Pearson correlation: calculate pmcc 5.08b Linear coding: effect on pmcc 5.09a Dependent/independent variables 5.09c Calculate regression line

5 As part of a study of charity shops in a small market town, two such shops, $X$ and $Y$, were each asked to provide details of its takings on 12 randomly selected days. The table shows, for each of the 12 days, the day's takings, $\pounds x$, of charity shop $X$ and the day's takings, $\pounds y$, of charity shop $Y$.

Day	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$	$\mathbf { K }$	$\mathbf { L }$
$\boldsymbol { x }$	46	57	39	116	62	77	41	61	15	53	68	61
$\boldsymbol { y }$	78	102	66	214	98	72	98	134	21	67	95	83

1. Calculate the value of the product moment correlation coefficient between $x$ and $y$.
2. Interpret your value in the context of this question.
Complete the scatter diagram shown on the opposite page.
The investigator realised subsequently that one of the 12 selected days was a particularly popular town market day and another was a day on which the weather was extremely severe. Identify each of these days giving a reason for each choice.
Removing the two days described in part (c) from the data gives the following information. $$S _ { x x } = 1292.5 \quad S _ { y y } = 3850.1 \quad S _ { x y } = 407.5$$
1. Use this information to recalculate the value of the product moment correlation coefficient between $x$ and $y$.
2. Hence revise, as necessary, your interpretation in part (a)(ii).
  [0pt] [3 marks] Shop $X$ takings(£) \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_33_21_294_1617}
  \end{figure} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_49_24_276_1710}
  \end{figure}
  \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_1304_415_406_1391}
  レ

Show mark scheme Show mark scheme source

Question 5:

Part (a)(i):

Answer	Marks	Guidance
Calculate $S_{xx}$, $S_{yy}$, $S_{xy}$ from raw data	M1	Correct method for sums
$r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$	M1	Correct formula used
$r \approx 0.986$	A1	Correct value

Part (a)(ii):

Answer	Marks	Guidance
Strong positive correlation between takings of shop $X$ and shop $Y$	B1	Mention of strong/positive
As takings of $X$ increase, takings of $Y$ increase	B1	Contextual interpretation

Part (b):

Answer	Marks	Guidance
Correct points plotted on scatter diagram	B1 B1	One mark per correct point

Part (c):

Answer	Marks	Guidance
Day D: market day (highest values for both $x$ and $y$, i.e. $x=116$, $y=214$)	B1	Correct identification
Day I: severe weather day (lowest values, $x=15$, $y=21$)	B1	Correct identification
Reason given for each	B1	Both reasons stated

Part (d)(i):

Answer	Marks	Guidance
$r = \frac{407.5}{\sqrt{1292.5 \times 3850.1}}$	M1	Correct substitution
$= \frac{407.5}{\sqrt{4976228}}$	M1	Correct calculation
$r \approx 0.578$	A1	Correct value

Part (d)(ii):

Answer	Marks	Guidance
Correlation is now only moderate/weak positive; removal of outliers has reduced the correlation significantly	B1	Revised interpretation in context

## Question 5:

### Part (a)(i):
Calculate $S_{xx}$, $S_{yy}$, $S_{xy}$ from raw data | M1 | Correct method for sums
$r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$ | M1 | Correct formula used
$r \approx 0.986$ | A1 | Correct value

### Part (a)(ii):
Strong positive correlation between takings of shop $X$ and shop $Y$ | B1 | Mention of strong/positive
As takings of $X$ increase, takings of $Y$ increase | B1 | Contextual interpretation

### Part (b):
Correct points plotted on scatter diagram | B1 B1 | One mark per correct point

### Part (c):
Day D: market day (highest values for both $x$ and $y$, i.e. $x=116$, $y=214$) | B1 | Correct identification
Day I: severe weather day (lowest values, $x=15$, $y=21$) | B1 | Correct identification
Reason given for each | B1 | Both reasons stated

### Part (d)(i):
$r = \frac{407.5}{\sqrt{1292.5 \times 3850.1}}$ | M1 | Correct substitution
$= \frac{407.5}{\sqrt{4976228}}$ | M1 | Correct calculation
$r \approx 0.578$ | A1 | Correct value

### Part (d)(ii):
Correlation is now only moderate/weak positive; removal of outliers has reduced the correlation significantly | B1 | Revised interpretation in context

Show LaTeX source

5 As part of a study of charity shops in a small market town, two such shops, $X$ and $Y$, were each asked to provide details of its takings on 12 randomly selected days.

The table shows, for each of the 12 days, the day's takings, $\pounds x$, of charity shop $X$ and the day's takings, $\pounds y$, of charity shop $Y$.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Day & $\mathbf { A }$ & $\mathbf { B }$ & $\mathbf { C }$ & $\mathbf { D }$ & $\mathbf { E }$ & $\mathbf { F }$ & $\mathbf { G }$ & $\mathbf { H }$ & $\mathbf { I }$ & $\mathbf { J }$ & $\mathbf { K }$ & $\mathbf { L }$ \\
\hline
$\boldsymbol { x }$ & 46 & 57 & 39 & 116 & 62 & 77 & 41 & 61 & 15 & 53 & 68 & 61 \\
\hline
$\boldsymbol { y }$ & 78 & 102 & 66 & 214 & 98 & 72 & 98 & 134 & 21 & 67 & 95 & 83 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the value of the product moment correlation coefficient between $x$ and $y$.
\item Interpret your value in the context of this question.
\end{enumerate}\item Complete the scatter diagram shown on the opposite page.
\item The investigator realised subsequently that one of the 12 selected days was a particularly popular town market day and another was a day on which the weather was extremely severe.

Identify each of these days giving a reason for each choice.
\item Removing the two days described in part (c) from the data gives the following information.

$$S _ { x x } = 1292.5 \quad S _ { y y } = 3850.1 \quad S _ { x y } = 407.5$$
\begin{enumerate}[label=(\roman*)]
\item Use this information to recalculate the value of the product moment correlation coefficient between $x$ and $y$.
\item Hence revise, as necessary, your interpretation in part (a)(ii).\\[0pt]
[3 marks]

Shop $X$ takings(£)

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{harity Shops}
  \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_33_21_294_1617}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{harity Shops}
  \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_49_24_276_1710}
\end{center}
\end{figure}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_1304_415_406_1391}
\end{center}

レ
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2014 Q5 [13]}}

This paper (7 questions)

View full paper

Q1 6 Q2 10 Q3 11 Q4 10 Q5 13 Q6 14 Q7 11

Day	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)	\(\mathbf { G }\)	\(\mathbf { H }\)	\(\mathbf { I }\)	\(\mathbf { J }\)	\(\mathbf { K }\)	\(\mathbf { L }\)
\(\boldsymbol { x }\)	46	57	39	116	62	77	41	61	15	53	68	61
\(\boldsymbol { y }\)	78	102	66	214	98	72	98	134	21	67	95	83

Answer	Marks	Guidance
Calculate \(S_{xx}\), \(S_{yy}\), \(S_{xy}\) from raw data	M1	Correct method for sums
\(r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}\)	M1	Correct formula used
\(r \approx 0.986\)	A1	Correct value

Answer	Marks	Guidance
Strong positive correlation between takings of shop \(X\) and shop \(Y\)	B1	Mention of strong/positive
As takings of \(X\) increase, takings of \(Y\) increase	B1	Contextual interpretation

Answer	Marks	Guidance
\(r = \frac{407.5}{\sqrt{1292.5 \times 3850.1}}\)	M1	Correct substitution
\(= \frac{407.5}{\sqrt{4976228}}\)	M1	Correct calculation
\(r \approx 0.578\)	A1	Correct value