| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sampling distribution theory |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing basic understanding of sampling distributions through enumeration. Parts (a)-(b) are definitional recall, part (c) is routine weighted mean/variance calculation, and parts (d)-(e) involve listing all possible samples (only 8 outcomes) and computing their probabilities—mechanical work with no novel insight required. Slightly easier than average due to small sample space and step-by-step scaffolding. |
| Spec | 2.01a Population and sample: terminology5.01a Permutations and combinations: evaluate probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05a Sample mean distribution: central limit theorem |
| VIIIV SIHI NI IIIYM ION OC | VIUV SIHI NI JIIIM I I ON OC | VEXV SIHII NI JIIIM I ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| A random variable that is a function of a (random) sample involving no unknown quantities/parameters; or a quantity calculated solely from a random sample | B1 | Requires: (1) function/quantity, (2) sample/observations/data, (3) no unknown parameters |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| If all possible samples are chosen from a population; then the values of a statistic and the associated probabilities is a sampling distribution; or a probability distribution of a statistic | B1 | Requires all underlined words: All values of a statistic with their associated probabilities; or probability distribution of a statistic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Mean \(= 100 \times \frac{4}{7} + 200 \times \frac{3}{7} = \frac{1000}{7}\) | B1 | awrt 143 |
| Variance \(= 100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} - \left(\frac{1000}{7}\right)^2\) | M1 | \(100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} -\) (their mean)\(^2\) |
| \(= \frac{120000}{49}\) | A1 | awrt 2450 (to 3sf) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((100,100,100)\) | B2 | Any 2 of the listed sample types for B1; all correct for B2 |
| \((100,100,200)\), \((100,200,100)\), \((200,100,100)\) or \(3\times(100,100,200)\) | Allow other notation for 100 and 200 e.g. Small and Large | |
| \((100,200,200)\), \((200,100,200)\), \((200,200,100)\) or \(3\times(100,200,200)\) | ||
| \((200,200,200)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((100,100,100)\): \(\left(\frac{4}{7}\right)^3 = \frac{64}{343}\) awrt 0.187; \((200,200,200)\): \(\left(\frac{3}{7}\right)^3 = \frac{27}{343}\) awrt 0.0787 | B1 | Both probabilities correct |
| \((100,100,200)\): \(3\times\left(\frac{4}{7}\right)^2\times\left(\frac{3}{7}\right) = \frac{144}{343}\) awrt 0.420 | M1 | \(3 \times p^2 \times (1-p)\) |
| \((100,200,200)\): \(3\times\left(\frac{4}{7}\right)\times\left(\frac{3}{7}\right)^2 = \frac{108}{343}\) awrt 0.315 | A1 | Either correct |
| Full table with \(m\): \(100\), \(\frac{400}{3}\), \(\frac{500}{3}\), \(200\) and all probabilities correct | A1 | All means correct and all probabilities correct (table not required but means must be associated with correct probabilities) |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| A random variable that is a function of a (random) sample involving no unknown quantities/parameters; or a quantity calculated solely from a random sample | B1 | Requires: (1) function/quantity, (2) sample/observations/data, (3) no unknown parameters |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| If all possible samples are chosen from a population; then the values of a statistic and the associated probabilities is a sampling distribution; or a probability distribution of a statistic | B1 | Requires all underlined words: **All values** of a **statistic** with their associated **probabilities**; or **probability distribution** of a **statistic** |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Mean $= 100 \times \frac{4}{7} + 200 \times \frac{3}{7} = \frac{1000}{7}$ | B1 | awrt 143 |
| Variance $= 100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} - \left(\frac{1000}{7}\right)^2$ | M1 | $100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} -$ (their mean)$^2$ |
| $= \frac{120000}{49}$ | A1 | awrt 2450 (to 3sf) |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(100,100,100)$ | B2 | Any 2 of the listed sample types for B1; all correct for B2 |
| $(100,100,200)$, $(100,200,100)$, $(200,100,100)$ or $3\times(100,100,200)$ | | Allow other notation for 100 and 200 e.g. Small and Large |
| $(100,200,200)$, $(200,100,200)$, $(200,200,100)$ or $3\times(100,200,200)$ | | |
| $(200,200,200)$ | | |
## Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(100,100,100)$: $\left(\frac{4}{7}\right)^3 = \frac{64}{343}$ awrt 0.187; $(200,200,200)$: $\left(\frac{3}{7}\right)^3 = \frac{27}{343}$ awrt 0.0787 | B1 | Both probabilities correct |
| $(100,100,200)$: $3\times\left(\frac{4}{7}\right)^2\times\left(\frac{3}{7}\right) = \frac{144}{343}$ awrt 0.420 | M1 | $3 \times p^2 \times (1-p)$ |
| $(100,200,200)$: $3\times\left(\frac{4}{7}\right)\times\left(\frac{3}{7}\right)^2 = \frac{108}{343}$ awrt 0.315 | A1 | Either correct |
| Full table with $m$: $100$, $\frac{400}{3}$, $\frac{500}{3}$, $200$ and all probabilities correct | A1 | All means correct **and** all probabilities correct (table not required but means must be associated with correct probabilities) |3. Explain what you understand by
\begin{enumerate}[label=(\alph*)]
\item a statistic,
\item a sampling distribution.
A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
\item Find the mean and the variance of the number of screws in the packets stored at the factory.
A random sample of 3 packets is taken from the factory and $Y _ { 1 } , Y _ { 2 }$ and $Y _ { 3 }$ denote the number of screws in each of these packets.
\item List all the possible samples.
\item Find the sampling distribution of $\bar { Y }$\\
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIIIV SIHI NI IIIYM ION OC & VIUV SIHI NI JIIIM I I ON OC & VEXV SIHII NI JIIIM I ION OO \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q3 [11]}}