| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Non-geometric distribution identification |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic understanding of statistics vs parameters, binomial distribution identification, and geometric probability calculation. Part (a) requires only definitional knowledge, part (b) is direct recognition of binomial conditions, and part (c) is a standard geometric probability formula application with given probability 0.4. All parts are routine recall/recognition with minimal problem-solving. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02f Geometric distribution: conditions5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (i) \(S\) is a statistic, (ii) \(D\) is not a statistic, (iii) \(F\) is a statistic | B1, B1, B1 (3) | For each variable. Accept "yes, no, yes" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(T \sim B(10, 0.4)\) | M1A1 (2) | M1 for binomial; A1 for \(n=10\) and \(p=0.4\). If 2 options given, must select correct one or gain M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(2' \ 2' \ 2)\) or \(P(5 \ 5 \ 2, \ 5{>}5 \ 2, \ {>}5{>}5 \ 2)\) | M1 | Identifying correct possibilities in correct order. Condone \(2\times(5{>}5\ 2)\) or \(2\times({>}5\ 5\ 2)\) if implied by correct answer |
| \(= 0.6^2 \times 0.4 = (0.25)^2(0.4) + 2\times(0.25)(0.35)(0.4) + (0.35)^2(0.4)\) | ||
| \(= 0.144\) | A1 (2) | For 0.144 or exact equivalent \(\frac{18}{125}\) |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (i) $S$ is a statistic, (ii) $D$ is not a statistic, (iii) $F$ is a statistic | B1, B1, B1 (3) | For each variable. Accept "yes, no, yes" |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $T \sim B(10, 0.4)$ | M1A1 (2) | M1 for binomial; A1 for $n=10$ and $p=0.4$. If 2 options given, must select correct one or gain M0A0 |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(2' \ 2' \ 2)$ or $P(5 \ 5 \ 2, \ 5{>}5 \ 2, \ {>}5{>}5 \ 2)$ | M1 | Identifying correct possibilities in correct order. Condone $2\times(5{>}5\ 2)$ or $2\times({>}5\ 5\ 2)$ if implied by correct answer |
| $= 0.6^2 \times 0.4 = (0.25)^2(0.4) + 2\times(0.25)(0.35)(0.4) + (0.35)^2(0.4)$ | | |
| $= 0.144$ | A1 (2) | For 0.144 or exact equivalent $\frac{18}{125}$ |
---2. A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter $\mu$. The number 2 is on $40 \%$ of the counters and the number 5 is on $25 \%$ of the counters. All the remaining counters have numbers greater than 5 on them.
A random sample of 10 counters is taken from the bag.
\begin{enumerate}[label=(\alph*)]
\item State whether or not each of the following is a statistic
\begin{enumerate}[label=(\roman*)]
\item $S =$ the sum of the numbers on the counters in the sample,
\item $D =$ the difference between the highest number in the sample and $\mu$,
\item $F =$ the number of counters in the sample with a number 5 on them.
The random variable $T$ represents the number of counters in a random sample of 10 with the number 2 on them.
\end{enumerate}\item Specify the sampling distribution of $T$.
The counters are selected one by one.
\item Find the probability that the third counter selected is the first counter with the number 2 on it.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2014 Q2 [7]}}