| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate from summary statistics |
| Difficulty | Moderate -0.8 This is a standard S1 linear regression question requiring routine application of formulas (Sₓᵧ, regression line coefficients) with all summary statistics provided. The only slight challenge is the interpretation questions (d) and (e), but these test basic understanding of interpolation vs extrapolation and sample population relevance—standard textbook concepts with minimal problem-solving required. |
| Spec | 2.02c Scatter diagrams and regression lines5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| \(f\) | 23 | 26 | 23 | 22 | 27 | 24 | 20 | 21 |
| \(h\) | 135 | 144 | 134 | 136 | 140 | 134 | 130 | 132 |
| Answer | Marks |
|---|---|
| \((S_{jh} =) 25291 - \frac{186 \times 1085}{8}\) | M1 |
| = 64.75 (accept 64.8) | A1 |
| Answer | Marks |
|---|---|
| \(b = \frac{"64.75"}{39.5}\) = 1.6392.... (awrt 1.6) | M1, A1 |
| \(a = \frac{1085}{8} - b \times \frac{186}{8}\) = 97.512... (awrt 97.5) | M1, A1 |
| \(h = 97.5 + 1.64f\) | A1ft (dep on M1M1) |
| Answer | Marks |
|---|---|
| \(h = 97.5 + 1.64 \times 25\) = 138−139 (final answer in [138, 139]) | M1, A1 |
| Answer | Marks |
|---|---|
| Should be reliable, since 25 cm(or f or footlength) is within the range of the data | B1, B1 |
| Answer | Marks |
|---|---|
| Line is for children – a different equation would apply to adults or Children are still growing, height will increase more than foot length | B1 |
## (a)
$(S_{jh} =) 25291 - \frac{186 \times 1085}{8}$ | M1 |
= 64.75 (accept 64.8) | A1 |
## (b)
$b = \frac{"64.75"}{39.5}$ = 1.6392.... (awrt 1.6) | M1, A1 |
$a = \frac{1085}{8} - b \times \frac{186}{8}$ = 97.512... (awrt 97.5) | M1, A1 |
$h = 97.5 + 1.64f$ | A1ft (dep on M1M1) |
## (c)
$h = 97.5 + 1.64 \times 25$ = 138−139 (final answer in [138, 139]) | M1, A1 |
## (d)
Should be reliable, since 25 cm(or f or footlength) is within the range of the data | B1, B1 |
## (e)
Line is for children – a different equation would apply to adults or Children are still growing, height will increase more than foot length | B1 |
**Notes:**
- **(a)** [NB r = 0.871 so do not confuse this with question 1] M1 for attempting a correct expression [allow a copying slip e.g. 25921]
- **(b)** 1st M1 for a correct expression for b, ft their part (a) but not $S_h = 25291$
- 1st A1 for awrt 1.6
- 2nd M1 for use of $a = \bar{h} - b \times \bar{f}$, ft their value for b. Must use $\bar{h}$ and $\bar{f}$ not values from table.
- 2nd A1 for awrt 97.5 [NB a = 135 − 1.63 × 23 = 97.51 but M0A0 since not using $\bar{h}$ and $\bar{f}$]
- 3rd A1ft for an equation for h and f with their coefficients to 3sf. **Dependent on both Ms**. Must be 3sf not awrt. Give this mark if seen in (c). Equation must be in h and f not y and x.
- **(c)** M1 for using their equation and f = 25 to find h
- A1 for their final answer in [138, 139]. Can give if they have 137.7... but round to 138
- **(d)** 1st B1 for suggesting it is reliable
- 2nd B1 for mentioning that 25 cm is within range of data. "interpolation"or"not extrapol"'B1. Use of "it" or a comment that height is in range is B0 but apply ISW
- **(e)** B1 for some comment that states a difference between children and teachers(adults). Must mention **teacher/adults** and **children**. e.g. ".teacher is not in same age group as the children", "equation is for children not adults" ".children and adults are different populations"", "teacher will be taller" is B0 since no mention of children. "equation is **only** valid for children" is OK since "only" implies not suitable for adults. **Or** Reference to different growth rates
---A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, $f$ cm, and their height, $h$ cm. The results are given in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$f$ & 23 & 26 & 23 & 22 & 27 & 24 & 20 & 21 \\
\hline
$h$ & 135 & 144 & 134 & 136 & 140 & 134 & 130 & 132 \\
\hline
\end{tabular}
\end{center}
(You may use $\sum f = 186 \quad \sum h = 1085 \quad S_{ff} = 39.5 \quad S_{hh} = 139.875 \quad \sum fh = 25291$)
\begin{enumerate}[label=(\alph*)]
\item Calculate $S_{fh}$ [2]
\item Find the equation of the regression line of $h$ on $f$ in the form $h = a + bf$.
Give the value of $a$ and the value of $b$ correct to 3 significant figures. [5]
\item Use your equation to estimate the height of a child with a left foot length of 25 cm. [2]
\item Comment on the reliability of your estimate in (c), giving a reason for your answer. [2]
\end{enumerate}
The left foot length of the teacher is 25 cm.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Give a reason why the equation in (b) should not be used to estimate the teacher's height. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2011 Q7 [12]}}