| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find parameter from normal approximation |
| Difficulty | Challenging +1.2 This requires setting up two simultaneous equations from normal approximation conditions (including continuity correction), then solving for n and p using inverse normal tables. It's more challenging than routine single-parameter problems but follows a standard S2 procedure with multiple computational steps. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(\frac{64.5 - \mu}{\sigma} = 0.75\) | B1 M1 M1 A1 | B1: \(\pm 0.75\) and \(\pm 1.25\) (or better) seen; 1st M1: \(64 \pm 0.5\) or \(52 \pm 0.5\); 2nd M1: standardising using \(64, 65\) or \(64\pm0.5\) or \(52, 53\) or \(52\pm0.5\) with \(\mu\) and \(\sigma\), or \(np\) and \(\sqrt{np(1-p)}\); 1st A1: \(\frac{64.5-\mu}{\sigma} = 0.75\) with compatible signs |
| \(\frac{52.5 - \mu}{\sigma} = -1.25\) | A1 | 2nd A1: \(\frac{52.5-\mu}{\sigma} = -1.25\) with compatible signs |
| \(64.5 - \mu = 0.75\sigma\) | dM1 | 3rd M1: solving simultaneous equations dependent on 2nd M1; must attempt to eliminate \(\mu\) or \(\sigma\), or \(np\) or \(\sqrt{np(1-p)}\) |
| \(52.5 - \mu = -1.25\sigma\) | ||
| \(\sigma = 6\) | A1 | 3rd A1 |
| \(\mu = 60\) | A1 | 4th A1 |
| \(np = 60\) | M1 | 4th M1: using \(\mu = np\) (may be awarded at any stage) |
| \(np(1-p) = 36\) | M1 | 5th M1: using \(\sigma = \sqrt{np(1-p)}\) (may be awarded at any stage) |
| \(1 - p = 0.6\) | ||
| \(p = 0.4\) | A1 | |
| \(n = 150\) | A1 | |
| Total | (12) |
## Question 7:
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\frac{64.5 - \mu}{\sigma} = 0.75$ | B1 M1 M1 A1 | B1: $\pm 0.75$ and $\pm 1.25$ (or better) seen; 1st M1: $64 \pm 0.5$ or $52 \pm 0.5$; 2nd M1: standardising using $64, 65$ or $64\pm0.5$ or $52, 53$ or $52\pm0.5$ with $\mu$ and $\sigma$, or $np$ and $\sqrt{np(1-p)}$; 1st A1: $\frac{64.5-\mu}{\sigma} = 0.75$ with compatible signs |
| $\frac{52.5 - \mu}{\sigma} = -1.25$ | A1 | 2nd A1: $\frac{52.5-\mu}{\sigma} = -1.25$ with compatible signs |
| $64.5 - \mu = 0.75\sigma$ | dM1 | 3rd M1: solving simultaneous equations dependent on 2nd M1; must attempt to eliminate $\mu$ or $\sigma$, or $np$ or $\sqrt{np(1-p)}$ |
| $52.5 - \mu = -1.25\sigma$ | | |
| $\sigma = 6$ | A1 | 3rd A1 |
| $\mu = 60$ | A1 | 4th A1 |
| $np = 60$ | M1 | 4th M1: using $\mu = np$ (may be awarded at any stage) |
| $np(1-p) = 36$ | M1 | 5th M1: using $\sigma = \sqrt{np(1-p)}$ (may be awarded at any stage) |
| $1 - p = 0.6$ | | |
| $p = 0.4$ | A1 | |
| $n = 150$ | A1 | |
| **Total** | **(12)** | |\begin{enumerate}
\item The random variable $Y \sim \mathrm {~B} ( n , p )$.
\end{enumerate}
Using a normal approximation the probability that $Y$ is at least 65 is 0.2266 and the probability that $Y$ is more than 52 is 0.8944
Find the value of $n$ and the value of $p$.\\
\hfill \mbox{\textit{Edexcel S2 2014 Q7 [12]}}