| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate regression line equation |
| Difficulty | Moderate -0.8 This is a standard S1 regression/correlation question requiring routine application of formulas with given summary statistics (Sbb, Smm). Parts (a)-(d) involve straightforward calculations using standard formulas, and part (e) tests basic understanding of how outliers affect correlation. No problem-solving or novel insight required—purely procedural work that any well-prepared S1 student should handle comfortably. |
| Spec | 2.02d Informal interpretation of correlation2.05f Pearson correlation coefficient |
| \(m\) | 29 | 29 | 35 | 39 | 41 | 43 | 44 | 46 |
| \(b\) | 75 | 83 | 91 | 121 | 120 | 126 | 119 | 126 |
| Answer | Marks | Guidance |
|---|---|---|
| \(29\times75 + 29\times83 + \ldots + 46\times126 = 33856\) | B1cao | for 33856 as final answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sum m = 306\) and \(\sum b = 861\); \(S_{bm} = 33856 - \frac{861\times306}{8} = 922.75\) | awrt 923; M1 A1 | B1 for both \(\sum m\) and \(\sum b\) seen or implied; M1 for correct formula ft answer to (a); A1 for awrt 923 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = \frac{922.75}{\sqrt{3083.875\times305.5}} = 0.9506706\ldots\) | awrt 0.951; M1 A1 | M1 for correct formula with \(S_{bm}\) and given \(S_{bb}\), \(S_{mm}\); A1 for awrt 0.951 |
| Answer | Marks | Guidance |
|---|---|---|
| As milk price increases, so does bread price | B1 | Must use words "milk" and "bread"; "as \(m\) increases \(b\) increases" is B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Since bread price increases but milk price stays the same, therefore the correlation will decrease (or be weaker) | B1, dB1 | 1st B1 for suitable reason e.g. \(m=46\), \(b=175\) is outlier; 2nd dB1 dep. on 1st B1 for saying \(r\) will decrease |
# Question 3:
## Part (a)
| $29\times75 + 29\times83 + \ldots + 46\times126 = 33856$ | B1cao | for 33856 as final answer |
## Part (b)
| $\sum m = 306$ and $\sum b = 861$; $S_{bm} = 33856 - \frac{861\times306}{8} = 922.75$ | awrt 923; M1 A1 | B1 for both $\sum m$ and $\sum b$ seen or implied; M1 for correct formula ft answer to (a); A1 for awrt 923 |
## Part (c)
| $r = \frac{922.75}{\sqrt{3083.875\times305.5}} = 0.9506706\ldots$ | awrt 0.951; M1 A1 | M1 for correct formula with $S_{bm}$ and given $S_{bb}$, $S_{mm}$; A1 for awrt 0.951 |
## Part (d)
| As milk price increases, so does bread price | B1 | Must use words "milk" and "bread"; "as $m$ increases $b$ increases" is B0 |
## Part (e)
| Since bread price increases but milk price stays the same, therefore the correlation will decrease (or be weaker) | B1, dB1 | 1st B1 for suitable reason e.g. $m=46$, $b=175$ is outlier; 2nd dB1 dep. on 1st B1 for saying $r$ will decrease |
---\begin{enumerate}
\item The table shows the price of a bottle of milk, $m$ pence, and the price of a loaf of bread, $b$ pence, for 8 different years.
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
$m$ & 29 & 29 & 35 & 39 & 41 & 43 & 44 & 46 \\
\hline
$b$ & 75 & 83 & 91 & 121 & 120 & 126 & 119 & 126 \\
\hline
\end{tabular}
\end{center}
(You may use $\mathrm { S } _ { b b } = 3083.875$ and $\mathrm { S } _ { m m } = 305.5$ )\\
(a) Find the exact value of $\sum b m$\\
(b) Find $\mathrm { S } _ { b m }$\\
(c) Calculate the product moment correlation coefficient between $b$ and $m$\\
(d) Interpret the value of the correlation coefficient.
A ninth year is added to the data set. In this year the price of the bottle of milk is 46 pence and the price of a loaf of bread is 175 pence.\\
(e) Without further calculation, state whether the value of the product moment correlation coefficient will increase, decrease or stay the same when all nine years are used. Give a reason for your answer.\\
\hfill \mbox{\textit{Edexcel S1 2015 Q3 [9]}}