| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Variance estimation from probability |
| Difficulty | Challenging +1.2 This is a multi-step CLT application requiring students to set up two simultaneous equations from probability statements, work backwards through z-scores, and solve for population parameters. While it requires understanding of sampling distributions and careful algebraic manipulation, it follows a standard S2 template with no novel conceptual leaps. The follow-up parts test theoretical understanding but are straightforward recall. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{\mu - 157.18}{\sigma/\sqrt{80}} = 1.282\); \(\dfrac{\mu - 164.76}{\sigma/\sqrt{80}} = 0.5244\) | M1 | Standardise once with \(\sqrt{80}\) or 80 and \(z\), signs may be wrong, allow "\(1-\)" errors. Allow cc, but *not* 0.1, 0.7, 0.9, 0.3 or \(\Phi(\text{these})\) \([= .5398,\ .758,\ .8159,\ .6179]\) |
| Both correct including signs, no cc | A1 | \(z\) may be wrong (provided it *is* \(z\)). Ignore signs |
| 1.28(155) seen anywhere, correct to 3 SF | B1 | Ignore signs |
| \([0.524,\ 0.525]\) seen anywhere | B1 | Ignore signs |
| Solve simultaneously: \(\mu = 170\) (169.98) | A1 | \(\mu\), a.r.t. 170 to 3 SF. CWO\(\times\)2 but allow from inaccurate \(z\) if answer(s) within limits |
| \(\sigma = 89.44\) | A1 | \(\sigma\), in range \([89, 90]\), *not* isw. *Don't* allow surds e.g. \(40\sqrt{5}\). \(-89.44\): A0A0 |
| Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (a) In using normal tables | B1 | Or equiv, e.g. "standardising", "dist of \(\bar{Y}\)". Any reference to \(\sigma/\sqrt{80}\): B0 |
| (b) Parent distribution not known | B1 | Allow "it is not normal", etc. No extras |
| (c) \(n\) large, nothing wrong seen [must be in correct order, no repeats] | B1 | If numerical, must be of the form "\(n > n_0\)" or "\(n \geq n_0\)" with \(30 \leq n_0 \leq 60\). *Not* "\(\geq 80\)" |
| Total: 3 |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{\mu - 157.18}{\sigma/\sqrt{80}} = 1.282$; $\dfrac{\mu - 164.76}{\sigma/\sqrt{80}} = 0.5244$ | M1 | Standardise once with $\sqrt{80}$ or 80 and $z$, signs may be wrong, allow "$1-$" errors. Allow cc, but *not* 0.1, 0.7, 0.9, 0.3 or $\Phi(\text{these})$ $[= .5398,\ .758,\ .8159,\ .6179]$ |
| Both correct **including signs**, no cc | A1 | $z$ may be wrong (provided it *is* $z$). Ignore signs |
| 1.28(155) seen anywhere, correct to 3 SF | B1 | Ignore signs |
| $[0.524,\ 0.525]$ seen anywhere | B1 | Ignore signs |
| Solve simultaneously: $\mu = 170$ (169.98) | A1 | $\mu$, a.r.t. 170 to 3 SF. CWO$\times$2 but allow from inaccurate $z$ if answer(s) within limits |
| $\sigma = 89.44$ | A1 | $\sigma$, in range $[89, 90]$, *not* isw. *Don't* allow surds e.g. $40\sqrt{5}$. $-89.44$: A0A0 |
| | **Total: 6** | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| **(a)** In using normal tables | B1 | Or equiv, e.g. "standardising", "dist of $\bar{Y}$". Any reference to $\sigma/\sqrt{80}$: B0 |
| **(b)** Parent distribution not known | B1 | Allow "it is not normal", etc. No extras |
| **(c)** $n$ large, nothing wrong seen [must be in correct order, no repeats] | B1 | If numerical, must be of the form "$n > n_0$" or "$n \geq n_0$" with $30 \leq n_0 \leq 60$. *Not* "$\geq 80$" |
| | **Total: 3** | |
---3 The mean of a sample of 80 independent observations of a continuous random variable $Y$ is denoted by $\bar { Y }$. It is given that $\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1$ and $\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7$.\\
(i) Calculate $\mathrm { E } ( Y )$ and the standard deviation of $Y$.\\
(ii) State
\begin{enumerate}[label=(\alph*)]
\item where in your calculations you have used the Central Limit Theorem,
\item why it was necessary to use the Central Limit Theorem,
\item why it was possible to use the Central Limit Theorem.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2013 Q3 [9]}}