| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.8 This is a straightforward S1 linear regression question requiring standard formula application: calculating Stm from given summaries, finding regression coefficients using bookwork formulas, and making a prediction. All steps are routine recall with no conceptual challenges or problem-solving required, making it easier than average but not trivial due to the arithmetic involved. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(t { } ^ { \circ } \mathrm { C }\) | 8 | 13 | 14 | 15 | 15 | 20 | 25 | 30 |
| \(m\) secs | 6.5 | 4.5 | 6 | 5 | 4 | 3 | 2 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\sum t=140\) (or \(\bar{t}=17.5\)) | B1 | For 140 seen in correct context or correctly labelled |
| \(\sum m=32\) (or \(\bar{m}=4\)) | B1 | For 32 seen in correct context or correctly labelled |
| \(S_{tm}=469.5-\frac{140\times 32}{8}\) | M1 | Attempting a correct expression; follow through their 140 and 32; may see \(\sum(t-\bar{t})(m-\bar{m})\) with all products |
| \(S_{tm}=-90.5\) | A1cso | Requires correct expression and no incorrect working leading to \(-90.5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(b=\frac{S_{tm}}{S_{tt}}=\frac{-90.5}{354}\) | M1 | Correct expression for \(b\); follow through their \(S_{tm}\); condone missing "\(-\)" |
| \(b=-0.255649\ldots\) \(\left(\text{allow }\frac{181}{708}\right)\), \(-0.25\) or awrt \(-0.26\) | A1 | |
| \(a=\frac{32}{8}-b\times\frac{140}{8}\) | M1 | Correct method for \(a\); follow through their sums from (a) and their \(b\) |
| \(m=8.47-0.256t\) \(\left(\text{allow }m=\frac{11999}{1416}-\frac{181}{708}t\right)\) | A1 | Correct equation in \(m\) and \(t\); \(a=\) awrt \(8.47\), \(b=\) awrt \(-0.256\); use of \(x\) or \(y\) scores A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((8.47-0.256\times 10=)\ 5.9\ldots\), awrt \(\mathbf{5.9}\) | B1 | Accept 6 if correct expression (awrt \(8.47-10\times\) awrt \(0.256\)) is seen |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Should be reliable since 10 is in the range (of the data) | B1 | Must mention 10 within the range; or "interpolating" / "not extrapolating"; "it is reliable since it is in the range" is B0 (not explicit enough) |
# Question 3:
## Part (a)
| Working | Mark | Guidance |
|---------|------|---------|
| $\sum t=140$ (or $\bar{t}=17.5$) | B1 | For 140 seen in correct context or correctly labelled |
| $\sum m=32$ (or $\bar{m}=4$) | B1 | For 32 seen in correct context or correctly labelled |
| $S_{tm}=469.5-\frac{140\times 32}{8}$ | M1 | Attempting a correct expression; follow through their 140 and 32; may see $\sum(t-\bar{t})(m-\bar{m})$ with all products |
| $S_{tm}=-90.5$ | A1cso | Requires correct expression and no incorrect working leading to $-90.5$ |
## Part (b)
| Working | Mark | Guidance |
|---------|------|---------|
| $b=\frac{S_{tm}}{S_{tt}}=\frac{-90.5}{354}$ | M1 | Correct expression for $b$; follow through their $S_{tm}$; condone missing "$-$" |
| $b=-0.255649\ldots$ $\left(\text{allow }\frac{181}{708}\right)$, $-0.25$ or awrt $-0.26$ | A1 | |
| $a=\frac{32}{8}-b\times\frac{140}{8}$ | M1 | Correct method for $a$; follow through their sums from (a) and their $b$ |
| $m=8.47-0.256t$ $\left(\text{allow }m=\frac{11999}{1416}-\frac{181}{708}t\right)$ | A1 | Correct equation in $m$ and $t$; $a=$ awrt $8.47$, $b=$ awrt $-0.256$; use of $x$ or $y$ scores A0 |
## Part (c)
| Working | Mark | Guidance |
|---------|------|---------|
| $(8.47-0.256\times 10=)\ 5.9\ldots$, awrt $\mathbf{5.9}$ | B1 | Accept 6 if correct expression (awrt $8.47-10\times$ awrt $0.256$) is seen |
## Part (d)
| Working | Mark | Guidance |
|---------|------|---------|
| Should be reliable since 10 is in the range (of the data) | B1 | Must mention 10 within the range; or "interpolating" / "not extrapolating"; "it is reliable since it is in the range" is B0 (not explicit enough) |
---3. A biologist is comparing the intervals ( $m$ seconds) between the mating calls of a certain species of tree frog and the surrounding temperature ( $t { } ^ { \circ } \mathrm { C }$ ). The following results were obtained.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$t { } ^ { \circ } \mathrm { C }$ & 8 & 13 & 14 & 15 & 15 & 20 & 25 & 30 \\
\hline
$m$ secs & 6.5 & 4.5 & 6 & 5 & 4 & 3 & 2 & 1 \\
\hline
\end{tabular}
\end{center}
$$\text { (You may use } \sum t m = 469.5 , \quad \mathrm {~S} _ { t t } = 354 , \quad \mathrm {~S} _ { m m } = 25.5 \text { ) }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { S } _ { t m } = - 90.5$
\item Find the equation of the regression line of $m$ on $t$ giving your answer in the form $m = a + b t$.
\item Use your regression line to estimate the time interval between mating calls when the surrounding temperature is $10 ^ { \circ } \mathrm { C }$.
\item Comment on the reliability of this estimate, giving a reason for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2013 Q3 [10]}}