| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Single probability inequality |
| Difficulty | Moderate -0.8 This is a straightforward application of the normal approximation to binomial distribution with standard steps: calculate mean and variance (simple formulas np and np(1-p)), apply continuity correction, and use normal tables. It's routine S2 content requiring only recall of the method with no problem-solving insight needed. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial |
| Answer | Marks |
|---|---|
| (a) Mean \(= 80 \times 0.375 = 30\), Variance \(= 80 \times 0.375 \times 0.625 = 18.75\) | M1 A1 M1 A1 |
| (b) \(X \sim B(80, 0.375) \approx N(30, 18.75)\) | M1 A1 |
| \(P(X > 40) = P(X > 40.5) = P\left(Z > \frac{10.5}{4.33}\right) = P(Z > 2.42)\) | M1 A1 A1 |
| \(= 1 - 0.9922 = 0.0078\) | M1 |
(a) Mean $= 80 \times 0.375 = 30$, Variance $= 80 \times 0.375 \times 0.625 = 18.75$ | M1 A1 M1 A1 |
(b) $X \sim B(80, 0.375) \approx N(30, 18.75)$ | M1 A1 |
$P(X > 40) = P(X > 40.5) = P\left(Z > \frac{10.5}{4.33}\right) = P(Z > 2.42)$ | M1 A1 A1 |
$= 1 - 0.9922 = 0.0078$ | M1 |
**Total: 11 marks**
---4. A random variable $X$ has the distribution $\mathrm { B } ( 80,0.375 )$.
\begin{enumerate}[label=(\alph*)]
\item Write down the mean and variance of $X$.
\item Use the Normal approximation to the binomial distribution to estimate $\mathrm { P } ( X > 40 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [11]}}