| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sampling distribution theory |
| Difficulty | Moderate -0.8 This is a straightforward S2 question testing basic definitions and enumeration of a small discrete sampling distribution. Parts (a)-(b) are pure recall, part (c) is routine expectation/variance calculation with a discrete distribution, and parts (d)-(e) require systematic listing of 7 cases and computing means—mechanical work with no conceptual challenge or novel insight required. |
| Spec | 2.01a Population and sample: terminology5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| A function of a (random) sample involving no unknown quantities/parameters or A quantity calculated solely from a random sample | B1 | Must include: (1) function/quantity/calculation, (2) sample/observations/data, (3) no unknown parameters |
| Answer | Marks | 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 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Mean} = 100 \times \frac{4}{7} + 200 \times \frac{3}{7} = \frac{1000}{7}\) | B1 | awrt 143 |
| \(\text{Variance} = 100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} - \left(\frac{1000}{7}\right)^2 = \frac{120000}{49}\) | M1, A1 | M1 for correct formula; A1 awrt 2450 (to 3sf) |
| Answer | Marks | Guidance |
|---|---|---|
| \((100,100,100)\); \((100,100,200)\), \((100,200,100)\), \((200,100,100)\) or \(3\times(100,100,200)\); \((100,200,200)\), \((200,100,200)\), \((200,200,100)\) or \(3\times(100,200,200)\); \((200,200,200)\) | B2 | B1 for any 2 correct; B1 all correct. Allow other notation e.g. Small and Large |
| Answer | Marks | 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 | Method \(3 \times p^2 \times (1-p)\); allow 0.42 |
| \((100,200,200)\): \(3\times\left(\frac{4}{7}\right)\times\left(\frac{3}{7}\right)^2 = \frac{108}{343}\) awrt 0.315 | A1 | Either correct |
| \(m\) | \(100\) | \(\frac{400}{3}\) |
| \(P(M=m)\) | \(\frac{64}{343}\) | \(\frac{144}{343}\) |
| Full correct table with means associated with correct probabilities | A1 | All means correct and all probabilities correct; table not required but means must be associated with correct probabilities |
## Question 3:
### Part (a):
| A function of a (random) sample involving no unknown quantities/parameters **or** A quantity calculated solely from a random sample | B1 | Must include: (1) function/quantity/calculation, (2) sample/observations/data, (3) no unknown parameters |
### Part (b):
| 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 |
### Part (c):
| $\text{Mean} = 100 \times \frac{4}{7} + 200 \times \frac{3}{7} = \frac{1000}{7}$ | B1 | awrt 143 |
| $\text{Variance} = 100^2 \times \frac{4}{7} + 200^2 \times \frac{3}{7} - \left(\frac{1000}{7}\right)^2 = \frac{120000}{49}$ | M1, A1 | M1 for correct formula; A1 awrt 2450 (to 3sf) |
### Part (d):
| $(100,100,100)$; $(100,100,200)$, $(100,200,100)$, $(200,100,100)$ or $3\times(100,100,200)$; $(100,200,200)$, $(200,100,200)$, $(200,200,100)$ or $3\times(100,200,200)$; $(200,200,200)$ | B2 | B1 for any 2 correct; B1 all correct. Allow other notation e.g. Small and Large |
### Part (e):
| $(100,100,100)$: $\left(\frac{4}{7}\right)^3 = \frac{64}{343}$ 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 | Method $3 \times p^2 \times (1-p)$; allow 0.42 |
| $(100,200,200)$: $3\times\left(\frac{4}{7}\right)\times\left(\frac{3}{7}\right)^2 = \frac{108}{343}$ awrt 0.315 | A1 | Either correct |
| $m$ | $100$ | $\frac{400}{3}$ | $\frac{500}{3}$ | $200$ |
|---|---|---|---|---|
| $P(M=m)$ | $\frac{64}{343}$ | $\frac{144}{343}$ | $\frac{108}{343}$ | $\frac{27}{343}$ |
| Full correct table with means associated with correct probabilities | 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 $\bar { Y }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2015 Q3 [11]}}