| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2001 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial distribution (parts a,b) and normal approximation to binomial (part c) with standard continuity correction. The calculations are routine for S2 level, requiring only direct application of learned techniques with no novel problem-solving. Part (d) is a standard bookwork assumption about independence. Slightly easier than average due to the mechanical nature of all parts. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X > 3) = 1 - P(X \leq 2) = 1 - 0.6778 = \underline{0.3222}\) or \(\underline{0.322}\) | M1, A1 | (2) |
| (b) \(P(X < 2) = P(X \leq 1) = \underline{0.3758}\) or \(\underline{0.376}\) | M1, A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(F \approx N(14, 11.2)\) | M1 (Normal approx), A1 \(\mu\), A1 \(\sigma^2\) | |
| \(P(F \leq 12) \approx P\!\left(z \leq \frac{12.5 - 14}{\sqrt{11.2}}\right)\) | \(\pm\frac{1}{2}\) M1, Standardising M1 | |
| \(= P(z \leq -0.4482\ldots)\), AWRT \(-0.45\) | A1 | |
| \(= 1 - 0.6736 = \underline{0.3264}\) | A1 | (AWRT \(0.326 \sim 0.327\)) (7) |
| (d) The 70 letters form a random sample or are representative, or letters are independent | B1 | (1) — (12) |
## Question 4:
$X$ = no. of letters marked 1st class, $X \sim B(10, 0.20)$
**(a)** $P(X > 3) = 1 - P(X \leq 2) = 1 - 0.6778 = \underline{0.3222}$ or $\underline{0.322}$ | M1, A1 | **(2)**
**(b)** $P(X < 2) = P(X \leq 1) = \underline{0.3758}$ or $\underline{0.376}$ | M1, A1 | **(2)**
**(c)** $F$ = no. of 1st class stamps in batch of 70, $F \sim B(70, 0.20)$
$F \approx N(14, 11.2)$ | M1 (Normal approx), A1 $\mu$, A1 $\sigma^2$ |
$P(F \leq 12) \approx P\!\left(z \leq \frac{12.5 - 14}{\sqrt{11.2}}\right)$ | $\pm\frac{1}{2}$ M1, Standardising M1 |
$= P(z \leq -0.4482\ldots)$, AWRT $-0.45$ | A1 |
$= 1 - 0.6736 = \underline{0.3264}$ | A1 | (AWRT $0.326 \sim 0.327$) **(7)**
**(d)** The 70 letters form a **random sample** or are **representative**, or letters are **independent** | B1 | **(1)** — **(12)**
---4. A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that $20 \%$ of letters to be posted are marked first class.
In a random selection of 10 letters to be posted, find the probability that the number marked first class is
\begin{enumerate}[label=(\alph*)]
\item at least 3,
\item fewer than 2 .
One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
\item use a suitable approximation to find the probability that there are enough first class stamps.
\item State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2001 Q4 [12]}}