| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | October |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate from summary statistics |
| Difficulty | Moderate -0.5 This is a standard S1 linear regression question requiring routine application of formulas for S_tt, S_ct, PMCC, and regression line calculations. All steps follow textbook procedures with no problem-solving insight needed, though the multi-part structure and final percentage comparison in part (f) add slight complexity beyond the most basic questions. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \([S_{tt}=]82873-\frac{1361^2}{40}\) and \([S_{ct}=]83634-\frac{1634\times 1361}{40}\) | M1 | Either correct expression |
| \([S_{tt}=]36564.975\) | A1 | 36 564.975 or exact equivalent |
| \([S_{ct}=]28037.15\) | A1 | 28037.15 or exact equivalent. SC M1A1A0 for awrt 36565 and awrt 28037 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \([r=]\frac{28037.15}{\sqrt{28732.1\times 36564.975}}=0.865\ldots\) | M1 | Valid attempt at \(r\) with their \(S_{cc}\neq 28732.1\) and their \(S_{ct}\neq 83634\) |
| awrt \(\mathbf{0.865}\) | A1 | awrt 0.865 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| In general, films with higher cost have higher ticket sales | B1ft | Only ft if \( |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \([b=]\frac{28037.15}{28732.1}[=0.9758\ldots]\) | M1 | Correct numerical expression for \(b\) ft their \(S_{ct}\neq 83634\). Implied by awrt 0.9758 or better |
| \([a=]\frac{1361}{40}-b\times\frac{1634}{40}\) or \(34.025-b\times 40.85\) | M1 | Attempt at \(a\) with their value of \(b\) substituted. Implied by awrt 5.837 or better |
| \(t=\text{awrt } {-5.84}+\text{awrt } 0.976c\) | A1cso* | Both \(b=0.9758\) (or better) or \(a=5.837\) (or better) must be seen, along with the correct equation in \(t\) and \(c\) (no fractions) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(t=-5.84+0.976\times 90\) | M1 | For substituting \(c=90\) into \(t=\text{awrt }{-5.84}+\text{awrt }0.976c\). Implied by awrt 82 |
| \(t=\text{£82 million}\) | A1 | £82 million (must include units). Allow awrt £82 million |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(-5.84+0.976c < 0.8c \rightarrow 0.176c < 5.84\) | M1 | Forming inequality (allow \(>, <, =,\) or …) with \(0.8c\) |
| \(c < \text{awrt (£) } \mathbf{33.2}\) | A1 | Correct inequality in \(c\) with awrt 33.2 (units not required). Do not allow as a fraction. Ignore any lower limit. Condone awrt 33200000 or awrt 33.2 million |
# Question 2:
## Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| $[S_{tt}=]82873-\frac{1361^2}{40}$ **and** $[S_{ct}=]83634-\frac{1634\times 1361}{40}$ | M1 | Either correct expression |
| $[S_{tt}=]36564.975$ | A1 | 36 564.975 or exact equivalent |
| $[S_{ct}=]28037.15$ | A1 | 28037.15 or exact equivalent. **SC** M1A1A0 for awrt 36565 and awrt 28037 |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| $[r=]\frac{28037.15}{\sqrt{28732.1\times 36564.975}}=0.865\ldots$ | M1 | Valid attempt at $r$ with their $S_{cc}\neq 28732.1$ and their $S_{ct}\neq 83634$ |
| awrt $\mathbf{0.865}$ | A1 | awrt 0.865 |
## Part (c)
| Working | Mark | Guidance |
|---------|------|----------|
| In general, films with higher **cost** have higher ticket **sales** | B1ft | Only ft if $|r|<1$. Must include underlined words (allow use of $c$ for cost and $t$ for sales). Must be compatible with value in (b). As one increases the other increases. Do not accept "$t$ and $c$ are similar." |
## Part (d)
| Working | Mark | Guidance |
|---------|------|----------|
| $[b=]\frac{28037.15}{28732.1}[=0.9758\ldots]$ | M1 | Correct numerical expression for $b$ ft their $S_{ct}\neq 83634$. Implied by awrt 0.9758 or better |
| $[a=]\frac{1361}{40}-b\times\frac{1634}{40}$ **or** $34.025-b\times 40.85$ | M1 | Attempt at $a$ with their value of $b$ substituted. Implied by awrt 5.837 or better |
| $t=\text{awrt } {-5.84}+\text{awrt } 0.976c$ | A1cso* | Both $b=0.9758$ (or better) **or** $a=5.837$ (or better) must be seen, along with the correct equation in $t$ and $c$ (no fractions) |
## Part (e)
| Working | Mark | Guidance |
|---------|------|----------|
| $t=-5.84+0.976\times 90$ | M1 | For substituting $c=90$ into $t=\text{awrt }{-5.84}+\text{awrt }0.976c$. Implied by awrt 82 |
| $t=\text{£82 million}$ | A1 | £82 million (must include units). Allow awrt £82 million |
## Part (f)
| Working | Mark | Guidance |
|---------|------|----------|
| $-5.84+0.976c < 0.8c \rightarrow 0.176c < 5.84$ | M1 | Forming inequality (allow $>, <, =,$ or …) with $0.8c$ |
| $c < \text{awrt (£) } \mathbf{33.2}$ | A1 | Correct inequality in $c$ with awrt 33.2 (units not required). Do not allow as a fraction. Ignore any lower limit. Condone awrt 33200000 or awrt 33.2 million |
---\begin{enumerate}
\item The production cost, $\pounds c$ million, of a film and the total ticket sales, $\pounds t$ million, earned by the film are recorded for a sample of 40 films.
\end{enumerate}
Some summary statistics are given below.
$$\sum c = 1634 \quad \sum t = 1361 \quad \sum t ^ { 2 } = 82873 \quad \sum c t = 83634 \quad \mathrm {~S} _ { c c } = 28732.1$$
(a) Find the exact value of $\mathrm { S } _ { t t }$ and the exact value of $\mathrm { S } _ { c t }$\\
(b) Calculate the value of the product moment correlation coefficient for these data.\\
(c) Give an interpretation of your answer to part (b)\\
(d) Show that the equation of the linear regression line of $t$ on $c$ can be written as
$$t = - 5.84 + 0.976 c$$
where the values of the intercept and gradient are given to 3 significant figures.\\
(e) Find the expected total ticket sales for a film with a production cost of $\pounds 90$ million.
Using the regression line in part (d)\\
(f) find the range of values of the production cost of a film for which the total ticket sales are less than $80 \%$ of its production cost.
\hfill \mbox{\textit{Edexcel S1 2022 Q2 [13]}}