| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2008 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Identify response/explanatory variables |
| Difficulty | Moderate -0.8 This is a routine S1 linear regression question requiring standard formula application (Stt, Svv, Stv, PMCC, regression line) with no conceptual challenges. All steps are algorithmic calculations following memorized procedures, making it easier than average for A-level. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_{tt} = 10922.81 - \dfrac{401.3^2}{15} = 186.6973\) | M1A1 | awrt 187; M1 any one correct formula attempt |
| \(S_{vv} = 42.3356 - \dfrac{25.08^2}{15} = 0.40184\) | A1 | awrt 0.402 |
| \(S_{tv} = 677.971 - \dfrac{401.3 \times 25.08}{15} = 6.9974\) | A1 | awrt 7.00 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = \dfrac{6.9974}{\sqrt{186.6973 \times 0.40184}}\) | M1A1ft | M1 for correct formula and attempt to use; A1ft from part (a) |
| \(= 0.807869\) | A1 | awrt 0.808; award 3 marks for awrt 0.808 with no working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(t\) is the explanatory variable as we can control temperature but not frequency of noise (or equivalent) | B1 B1 | Marks independent; second B1 requires interpretation in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| High value of \(r\) or \(r\) close to 1, or strong correlation | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(b = \dfrac{6.9974}{186.6973} = 0.03748\) | M1A1 | awrt 0.0375; M1 for values the right way up |
| \(a = \dfrac{25.08}{15} - b \times \dfrac{401.3}{15} = 0.6692874\) | M1A1 | awrt 0.669 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(t=19\): \(v = 0.6692874 + 0.03748 \times 19 = 1.381406\) | B1 | awrt 1.4 |
## Question 4:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_{tt} = 10922.81 - \dfrac{401.3^2}{15} = 186.6973$ | M1A1 | awrt 187; M1 any one correct formula attempt |
| $S_{vv} = 42.3356 - \dfrac{25.08^2}{15} = 0.40184$ | A1 | awrt 0.402 |
| $S_{tv} = 677.971 - \dfrac{401.3 \times 25.08}{15} = 6.9974$ | A1 | awrt 7.00 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \dfrac{6.9974}{\sqrt{186.6973 \times 0.40184}}$ | M1A1ft | M1 for correct formula and attempt to use; A1ft from part (a) |
| $= 0.807869$ | A1 | awrt 0.808; award 3 marks for awrt 0.808 with no working |
### Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $t$ is the explanatory variable as we can control temperature but not frequency of noise (or equivalent) | B1 B1 | Marks independent; second B1 requires interpretation in context |
### Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| High value of $r$ or $r$ close to 1, or strong correlation | B1 | |
### Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b = \dfrac{6.9974}{186.6973} = 0.03748$ | M1A1 | awrt 0.0375; M1 for values the right way up |
| $a = \dfrac{25.08}{15} - b \times \dfrac{401.3}{15} = 0.6692874$ | M1A1 | awrt 0.669 |
### Part (f)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $t=19$: $v = 0.6692874 + 0.03748 \times 19 = 1.381406$ | B1 | awrt 1.4 |
---4. Crickets make a noise. The pitch, $v \mathrm { kHz }$, of the noise made by a cricket was recorded at 15 different temperatures, $t ^ { \circ } \mathrm { C }$. These data are summarised below.
$$\sum t ^ { 2 } = 10922.81 , \sum v ^ { 2 } = 42.3356 , \sum t v = 677.971 , \sum t = 401.3 , \sum v = 25.08$$
\begin{enumerate}[label=(\alph*)]
\item Find $S _ { t t } , S _ { v v }$ and $S _ { t v }$ for these data.
\item Find the product moment correlation coefficient between $t$ and $v$.
\item State, with a reason, which variable is the explanatory variable.
\item Give a reason to support fitting a regression model of the form $v = a + b t$ to these data.
\item Find the value of $a$ and the value of $b$. Give your answers to 3 significant figures.
\item Using this model, predict the pitch of the noise at $19 ^ { \circ } \mathrm { C }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2008 Q4 [15]}}