| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Estimate from percentile/frequency data |
| Difficulty | Standard +0.3 This is a standard S2 question requiring students to set up equations using inverse normal distribution from given frequencies, then solve simultaneous equations. While it involves multiple steps (converting frequencies to probabilities, using z-tables/inverse normal, solving simultaneous equations), these are routine techniques practiced extensively in S2. The method is algorithmic rather than requiring insight, making it slightly easier than average overall A-level difficulty. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Interval | \(G < 40.0\) | \(40.0 \leqslant G < 60.0\) | \(G \geqslant 60.0\) |
| Frequency | 17 | 58 | 25 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{\mu - 40}{\sigma} = 0.9544\) | M1 | Standardise with \(\mu\) and \(\sigma\) and equate to \(\Phi^{-1}\); allow \(\sigma^2\) but not \(\sqrt{n}\); allow \(1-\), cc, wrong signs. \(P(\ldots)\): M0 here. But can recover both marks from part (ii). |
| B1 | \([0.954, 0.955]\) seen | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{60 - \mu}{\sigma} = 0.674(5)\) | M1 | Standardise as in (i) but do not give if "\(1-\)" or wrong signs in *either* equation |
| B1 | \([0.674, 0.675]\) seen. (Other errors lead to loss of A marks.) | |
| Solve to get \(\sigma = 12.3\) \([12.278]\) | A1 | \(\sigma\), a.r.t. 12.3, cwo |
| \(\mu = 51.7(18)\) | A1 | \(\mu\), a.r.t. 51.7, cwo [NB: *CARE!* either or both can be obtained from wrong equations.] {note for scoris zoning – (i) to be visible in marking (ii)} |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Based on a sample/small sample, etc | B1 | Any similar comment e.g. "frequencies not probabilities" (but not *just* "\(n\) is small") *and* no wrong comments. Not "because data is grouped". No scattergun. |
| [1] |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{\mu - 40}{\sigma} = 0.9544$ | M1 | Standardise with $\mu$ and $\sigma$ and equate to $\Phi^{-1}$; allow $\sigma^2$ but not $\sqrt{n}$; allow $1-$, cc, wrong signs. $P(\ldots)$: M0 here. But can recover both marks from part (ii). |
| | B1 | $[0.954, 0.955]$ seen |
| | **[2]** | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{60 - \mu}{\sigma} = 0.674(5)$ | M1 | Standardise as in (i) but do not give if "$1-$" or wrong signs in *either* equation |
| | B1 | $[0.674, 0.675]$ seen. (Other errors lead to loss of A marks.) |
| Solve to get $\sigma = 12.3$ $[12.278]$ | A1 | $\sigma$, a.r.t. 12.3, cwo |
| $\mu = 51.7(18)$ | A1 | $\mu$, a.r.t. 51.7, cwo [NB: ***CARE!*** either or both can be obtained from wrong equations.] {note for scoris zoning – (i) to be visible in marking (ii)} |
| | **[4]** | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Based on a sample/small sample, etc | B1 | Any similar comment e.g. "frequencies not probabilities" (but not *just* "$n$ is small") *and* no wrong comments. Not "because data is grouped". No scattergun. |
| | **[1]** | |
---3 The random variable $G$ has the distribution $\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)$. One hundred observations of $G$ are taken. The results are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
Interval & $G < 40.0$ & $40.0 \leqslant G < 60.0$ & $G \geqslant 60.0$ \\
\hline
Frequency & 17 & 58 & 25 \\
\hline
\end{tabular}
\end{center}
(i) By considering $\mathrm { P } ( G < 40.0 )$, write down an equation involving $\mu$ and $\sigma$. [2]\\
(ii) Find a second equation involving $\mu$ and $\sigma$. Hence calculate values for $\mu$ and $\sigma$. [4]\\[0pt]
(iii) Explain why your answers are only estimates. [1]
\hfill \mbox{\textit{OCR S2 2014 Q3 [7]}}