| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate summary statistics (Sxx, Syy, Sxy) |
| Difficulty | Moderate -0.8 This is a routine S1 statistics question testing standard formulas for Sxx, Sxy, correlation coefficient, and regression line. All calculations follow directly from memorized formulas with no problem-solving or interpretation challenges beyond basic substitution. Part (f) requires minimal conceptual understanding of outliers. Significantly easier than average A-level maths questions. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(S_H = 2688 - \frac{158^2}{10} = 191.6\) | M1 A1 | awrt 192; A1 awrt -5.03 |
| \(S_{TW} = 1760.62 - \frac{158 \times 111.75}{10} = -5.03\) | A1 | |
| (3 marks) | ||
| (b) \(r = \frac{-5.03}{\sqrt{191.6 \times 0.16}} = -0.908469\ldots\) | M1 A1 | awrt -0.908(5); M1 for correct attempt at use of formula, square root required. A1 awrt -0.908(5) |
| (2 marks) | ||
| (c) \(b = \frac{-5.03}{191.6} = -0.0263\) | M1 A1 | awrt -0.026 |
| \(a = 11.175 + 0.0263 \times 15.8\) | M1 | M1 for use of correct formula with \(b\) or 'their \(b\)'; require \(--\) or \(+\) and values in the correct place. |
| \(= 11.59\) | ||
| \(w = 11.6 - 0.0263t\) | A1 | A1 for equation as written with values awrt 3 sf, with \(w\) and \(t\). Accept fractional answers that are accurate to 3sf when evaluated as decimals |
| (4 marks) | ||
| (d) The explanatory variable is the age of each coin. This is because the age is set and the weight varies. | B1 B1 | B1 for 'Age' or \(t\) or 'years'; B1 for explanation |
| (2 marks) | ||
| (e) (i) awrt 11.5 | B1 | |
| (ii) | Decrease(in weight of coin of 0.1052 g) \(= 0.1\) or \(-0.1\) or increase of \(-0.1\) awrt(-0.1) | B1 |
| (2 marks) | ||
| (f) Decrease; removing the fake will result in a better linear fit so \(r\) will be closer to -1 | B1; B1 | B1 for Decrease only but 'mod \(r\) increases' explicitly stated in words or symbols award B1. B1 accept 'stronger correlation' or 'increase in correlation' or 'better linear fit' or ' \(r\) closer to -1' or 'points are closer to a straight line' or 'point is an outlier' or equivalent |
| (2 marks) |
**(a)** $S_H = 2688 - \frac{158^2}{10} = 191.6$ | M1 A1 | awrt 192; A1 awrt -5.03
| | |
| $S_{TW} = 1760.62 - \frac{158 \times 111.75}{10} = -5.03$ | A1 | |
| | (3 marks) |
**(b)** $r = \frac{-5.03}{\sqrt{191.6 \times 0.16}} = -0.908469\ldots$ | M1 A1 | awrt -0.908(5); M1 for correct attempt at use of formula, square root required. A1 awrt -0.908(5)
| | (2 marks) |
**(c)** $b = \frac{-5.03}{191.6} = -0.0263$ | M1 A1 | awrt -0.026
| | |
| $a = 11.175 + 0.0263 \times 15.8$ | M1 | M1 for use of correct formula with $b$ or 'their $b$'; require $--$ or $+$ and values in the correct place.
| | |
| $= 11.59$ | | |
| | |
| $w = 11.6 - 0.0263t$ | A1 | A1 for equation as written with values awrt 3 sf, with $w$ and $t$. Accept fractional answers that are accurate to 3sf when evaluated as decimals
| | (4 marks) |
**(d)** The explanatory variable is the age of each coin. This is because the age is set and the weight varies. | B1 B1 | B1 for 'Age' or $t$ or 'years'; B1 for explanation
| | (2 marks) |
**(e) (i)** awrt 11.5 | B1 | |
| |(ii)| Decrease(in weight of coin of 0.1052 g) $= 0.1$ or $-0.1$ or increase of $-0.1$ awrt(-0.1) | B1 | |
| | (2 marks) |
**(f)** Decrease; removing the fake will result in a better linear fit so $r$ will be closer to -1 | B1; B1 | B1 for Decrease only but 'mod $r$ increases' explicitly stated in words or symbols award B1. B1 accept 'stronger correlation' or 'increase in correlation' or 'better linear fit' or ' $r$ closer to -1' or 'points are closer to a straight line' or 'point is an outlier' or equivalent
| | (2 marks) |
**Total: 15 marks**
---\begin{enumerate}
\item The age, $t$ years, and weight, $w$ grams, of each of 10 coins were recorded. These data are summarised below.
\end{enumerate}
$$\sum t ^ { 2 } = 2688 \quad \sum t w = 1760.62 \quad \sum t = 158 \quad \sum w = 111.75 \quad S _ { w w } = 0.16$$
(a) Find $S _ { t t }$ and $S _ { t w }$ for these data.\\
(b) Calculate, to 3 significant figures, the product moment correlation coefficient between $t$ and $w$.\\
(c) Find the equation of the regression line of $w$ on $t$ in the form $w = a + b t$\\
(d) State, with a reason, which variable is the explanatory variable.\\
(e) Using this model, estimate\\
(i) the weight of a coin which is 5 years old,\\
(ii) the effect of an increase of 4 years in age on the weight of a coin.
It was discovered that a coin in the original sample, which was 5 years old and weighed 20 grams, was a fake.\\
(f) State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient. Give a reason for your answer.\\
\hfill \mbox{\textit{Edexcel S1 2012 Q5 [15]}}