| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Small sample binomial probability |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic concepts: identifying continuous variables (trivial), describing sampling (routine), and applying normal approximation to binomial with continuity correction (standard textbook procedure). The calculation in part (c) requires X ~ B(36, 1/3), approximate with N(12, 8), find P(X ≥ 20) using continuity correction—all mechanical steps with no problem-solving insight needed. |
| Spec | 2.01a Population and sample: terminology2.01c Sampling techniques: simple random, opportunity, etc2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\) = no. of competitors with \(A = 2\); \(X \sim B(36, \frac{1}{3})\) | M1 A1 | |
| \(X \approx \sim N(12, 8)\) | ||
| \(P(X \geq 20) \approx P\left(Z \geq \frac{19.5 - 12}{\sqrt{8}}\right)\) | M1, M1 | \(\pm\frac{1}{2}\), '\(z\)' |
| \(= P(Z \geq 2.65\ldots)\) | A1 | |
| \(= 1 - 0.9960 = 0.004\) | A1 | (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Probability is very low, so assumption of \(P(A=2) = \frac{1}{3}\) is unlikely. (Suggests \(P(A=2)\) might be higher.) | B1, B1 | (2) |
## Question 3 (continued):
### Part (c):
| $X$ = no. of competitors with $A = 2$; $X \sim B(36, \frac{1}{3})$ | M1 A1 | |
|---|---|---|
| $X \approx \sim N(12, 8)$ | | |
| $P(X \geq 20) \approx P\left(Z \geq \frac{19.5 - 12}{\sqrt{8}}\right)$ | M1, M1 | $\pm\frac{1}{2}$, '$z$' |
| $= P(Z \geq 2.65\ldots)$ | A1 | |
| $= 1 - 0.9960 = 0.004$ | A1 | (6) |
### Part (d):
| Probability is very low, so assumption of $P(A=2) = \frac{1}{3}$ is unlikely. (Suggests $P(A=2)$ might be higher.) | B1, B1 | (2) |
|---|---|---|
---3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable $D$ represents the greatest distance thrown by each competitor and the random variable $A$ represents the number of the attempt in which the competitor achieved their greatest distance.
\begin{enumerate}[label=(\alph*)]
\item State which of the two random variables $D$ or $A$ is continuous.
A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
\item Specify a suitable sampling frame and explain how such a sample could be taken.\\
(2 marks)\\
The coach assumes that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$, and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt.
Using a suitable approximation, and assuming that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$,
\item find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.\\
(6 marks)
\item Comment on the assumption that $\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [11]}}