| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Convert regression equation between coded and original |
| Difficulty | Moderate -0.8 This is a routine S1 regression question requiring standard formula application (b = S_xy/S_xx, a = ȳ - bx̄) and algebraic substitution to convert between coded and original variables. The steps are mechanical and well-practiced, making it easier than average A-level material. |
| Spec | 2.04a Discrete probability distributions5.09c Calculate regression line5.09d Linear coding: effect on regression |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(b = \frac{18.35}{312.1} = 0.058795...\), \(a = 5.8 - "0.058795..." \times 4.8 = \text{awrt } 5.52\), So \(y = 5.52 + 0.0588x\) | M1, M1, A1, A1 | (4) |
| (b) \(\frac{e}{10} = "5.52" + "0.0588" \times \left(\frac{g-60}{4}\right)\), \(4e = 220.71 + 0.588(g-60)\), \(e = 46 + 0.15g\) | M1, dM1, A1, A1 | (4) |
| (c) \(e = "46" + "0.15" \times 100 = 61\) | M1, A1 | (2) |
**(a)** $b = \frac{18.35}{312.1} = 0.058795...$, $a = 5.8 - "0.058795..." \times 4.8 = \text{awrt } 5.52$, So $y = 5.52 + 0.0588x$ | M1, M1, A1, A1 | (4) | 1st M1 for correct expression for $b$; 2nd M1 for correct expression for $a$ – fit their value of $b$; 1st A1 for $a = \text{awrt } 5.52$; 2nd A1 for correct equation in $y$ and $x$ with $a$ and $b$ correct to awrt 3 sf
**(b)** $\frac{e}{10} = "5.52" + "0.0588" \times \left(\frac{g-60}{4}\right)$, $4e = 220.71 + 0.588(g-60)$, $e = 46 + 0.15g$ | M1, dM1, A1, A1 | (4) | 1st M1 for substitutions into their equation to get equation in $e$ and $g$. Need $y = \frac{e}{10}$ and $x = \frac{g-60}{4}$; 2nd dM1 Dep. on 1st M1 for attempt to simplify (at least removing fractions). Allow one slip. 1st A1 for equation $e = \text{awrt } 46 + ...$; 2nd A1 for equation $e = \text{awrt } 46 + \text{awrt } 0.15g$
**(c)** $e = "46" + "0.15" \times 100 = 61$ | M1, A1 | (2) | M1 for substituting $g = 100$ into their new equation (or $x = 10$ and then attempting to $\times$ ans. by 10); A1 for awrt 61
**ALT** 1st M1 for use of $d = \frac{10 \times "their b"}{"}$ or sight of 0.15 used as gradient; 2nd dM1 Dep. on 1st M1 for use of $\bar{e} = 10 \times "their $\bar{y}$" or sight of 58 and use of $\bar{g} = 4 \times "their $\bar{x}$" + 60$ or sight of 79.2 and use of these values to find $c$ in $c = \bar{e} - d\bar{g}$
---\begin{enumerate}
\item Sammy is studying the number of units of gas, $g$, and the number of units of electricity, $e$, used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that $x = \frac { g - 60 } { 4 }$ and $y = \frac { e } { 10 }$. The results for the coded data are summarised below
\end{enumerate}
$$\sum x = 48.0 \quad \sum y = 58.0 \quad \mathrm {~S} _ { x x } = 312.1 \quad \mathrm {~S} _ { y y } = 2.10 \quad \mathrm {~S} _ { x y } = 18.35$$
(a) Find the equation of the regression line of $y$ on $x$ in the form $y = a + b x$.
Give the values of $a$ and $b$ correct to 3 significant figures.\\
(b) Hence find the equation of the regression line of $e$ on $g$ in the form $e = c + d g$.
Give the values of $c$ and $d$ correct to 2 significant figures.\\
(c) Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used.\\
(a)Find the probability distribution of $X$ .\\
(b)Write down the value of $\mathrm { F } ( 1.8 )$ .\\
(a)Find the probability distribution of $X$ .勤
\hfill \mbox{\textit{Edexcel S1 2013 Q1 [10]}}