| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Compare approximation methods |
| Difficulty | Standard +0.3 This is a straightforward application of standard S2 techniques: part (a) uses binomial directly, parts (b) and (c) apply normal approximation with continuity correction. Part (d) tests understanding of when Poisson approximation is needed (small p, large n). All steps are routine textbook exercises requiring recall of conditions and standard procedures rather than problem-solving or insight. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks |
|---|---|
| (a) No. of Cons \(\sim B(10, 0.35)\), so \(P(X \geq 2) = 1 - 0.086 = 0.914\) | M1 A1 A1 |
| (b) No. of Cons on MRL \(\sim B(500, 0.37) \approx N(185, 116.55)\), so | M1 A1 |
| \(P(X > 200) = P(X > 199.5) = P\left(Z > \frac{14.5}{10.79}\right) = P(Z > 1.34)\) | M1 M1 |
| \(= 1 - 0.9099 = 0.0901\) | A1 A1 |
| (c) No. of MRL \(\sim B(200, 0.02) \approx \text{Po}(4)\) | M1 A1 |
| so \(P(X \geq 5) = 1 - 0.6288 = 0.371\) | M1 A1 |
| (d) Binomial to Normal needs continuity correction, going from a discrete to a continuous distribution | B1 B1 |
(a) No. of Cons $\sim B(10, 0.35)$, so $P(X \geq 2) = 1 - 0.086 = 0.914$ | M1 A1 A1 |
(b) No. of Cons on MRL $\sim B(500, 0.37) \approx N(185, 116.55)$, so | M1 A1 |
$P(X > 200) = P(X > 199.5) = P\left(Z > \frac{14.5}{10.79}\right) = P(Z > 1.34)$ | M1 M1 |
$= 1 - 0.9099 = 0.0901$ | A1 A1 |
(c) No. of MRL $\sim B(200, 0.02) \approx \text{Po}(4)$ | M1 A1 |
so $P(X \geq 5) = 1 - 0.6288 = 0.371$ | M1 A1 |
(d) Binomial to Normal needs continuity correction, going from a discrete to a continuous distribution | B1 B1 |
**Total: 16 marks**
---6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was $35 \%$, and the percentage who voted for the Monster Raving Loony party was $2 \%$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that a random sample of 10 electors includes at least two Conservative voters.
Use suitable approximations to find
\item the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
\item the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
\item One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [16]}}