| Exam Board | Edexcel |
|---|---|
| Module | FS2 AS (Further Statistics 2 AS) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate regression line then predict |
| Difficulty | Standard +0.3 This is a straightforward application of standard regression formulas from the formula booklet. Parts (a)-(c) involve direct substitution into given formulas for correlation coefficient and regression line. Parts (d)-(e) require reading a residual plot and making basic interpretations. While it's a multi-part question worth several marks, each step is routine for Further Statistics students with no novel problem-solving required. Slightly easier than average A-level due to the mechanical nature of the calculations. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(S_{ww}=11\,386\,134 - 27(628.59)^2 [=717\,748.5213]\) | B1 | Correct expression for \(S_{ww}\) (implied by correct answer) |
| \(r=\frac{13082.3}{\sqrt{260.8\times 717\,748.5213}}\) | M1 | Complete method to find \(r\); use of \(S_{ww}=11\,386\,134\) is M0 |
| \(r=0.95618\ldots\) awrt 0.956 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Since \(r\) is close to 1, data is consistent with a linear model. | B1 | Correct explanation and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(b=\frac{13082.3}{260.8}[=50.162\ldots]\) | M1 | Setting up linear model by finding gradient |
| \(a=628.59 - {'}b{'} (31.07)\) | M1 | Attempting \(y\)-intercept of linear model |
| \(w=-930+50.2x\) | A1 | Correct model with \(b=\) awrt 50.2 and \(a=\) awrt \(-930\) (must use \(w\) and \(x\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(w=-930+50.2(32)+80\) | M1 | Using the model with the residual; allow \(\pm80\) |
| \(w=756.4\) | A1 | awrt 756 (allow awrt 755 from use of exact values) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Negative residuals for all 5 observations with \(x>33\) suggests the model systematically overestimates weights for the longest bream. | B1 | Must reference both the residuals and the model; negative correlation between residuals and length is B0 |
## Question 3:
### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $S_{ww}=11\,386\,134 - 27(628.59)^2 [=717\,748.5213]$ | B1 | Correct expression for $S_{ww}$ (implied by correct answer) |
| $r=\frac{13082.3}{\sqrt{260.8\times 717\,748.5213}}$ | M1 | Complete method to find $r$; use of $S_{ww}=11\,386\,134$ is M0 |
| $r=0.95618\ldots$ awrt **0.956** | A1 | |
**(3 marks)**
### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Since $r$ is close to 1, data is consistent with a linear model. | B1 | Correct explanation and conclusion |
**(1 mark)**
### Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $b=\frac{13082.3}{260.8}[=50.162\ldots]$ | M1 | Setting up linear model by finding gradient |
| $a=628.59 - {'}b{'} (31.07)$ | M1 | Attempting $y$-intercept of linear model |
| $w=-930+50.2x$ | A1 | Correct model with $b=$ awrt 50.2 and $a=$ awrt $-930$ (must use $w$ and $x$) |
**(3 marks)**
### Part (d):
| Working | Mark | Guidance |
|---------|------|----------|
| $w=-930+50.2(32)+80$ | M1 | Using the model with the residual; allow $\pm80$ |
| $w=756.4$ | A1 | awrt 756 (allow awrt 755 from use of exact values) |
**(2 marks)**
### Part (e):
| Working | Mark | Guidance |
|---------|------|----------|
| Negative residuals for all 5 observations with $x>33$ suggests the model systematically overestimates weights for the longest bream. | B1 | Must reference both the residuals and the model; negative correlation between residuals and length is B0 |
**(1 mark)**
---\begin{enumerate}
\item Gabriela is investigating a particular type of fish, called bream. She wants to create a model to predict the weight, $w$ grams, of bream based on their length, $x \mathrm {~cm}$.
\end{enumerate}
For a sample of 27 bream, some summary statistics are given below.
$$\begin{gathered}
\bar { x } = 31.07 \quad \bar { w } = 628.59 \quad \sum w ^ { 2 } = 11386134 \\
\mathrm {~S} _ { x w } = 13082.3 \quad \mathrm {~S} _ { x x } = 260.8
\end{gathered}$$
(a) Find the value of the product moment correlation coefficient between $x$ and $w$\\
(b) Explain whether the answer to part (a) is consistent with a linear model for these data.\\
(c) Find the equation of the regression line of $w$ on $x$ in the form $w = a + b x$
A residual plot for these data is shown below.\\
\includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-06_931_1790_1107_139}
One of the bream in the sample has a length of 32 cm .\\
(d) Find its weight.\\
(e) With reference to the residual plot, comment on the model for bream with lengths above 33 cm .
\hfill \mbox{\textit{Edexcel FS2 AS 2022 Q3 [10]}}