| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Statistics vs non-statistics identification |
| Difficulty | Moderate -0.8 Parts (a)-(c) are basic sampling theory requiring simple recall. Part (d) involves setting up probability equations from a two-draw scenario with replacement, requiring systematic enumeration of cases and solving simultaneous equations, but follows a standard S2 pattern with straightforward algebra once the cases are identified. |
| Spec | 2.01a Population and sample: terminology2.01c Sampling techniques: simple random, opportunity, etc5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| VIHV SIHII NI I IIIM I ON OC | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Taking a random sample is quicker/cheaper/easier (compared to asking all of the youth club members) | B1 | Any one of the given reasons. Ignore extraneous non-contradictory reasons |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| A list/register/database of all the youth club members | B1 | Idea of list(oe). Need all (oe) (e.g. complete list) and members |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The members | B1 | The members/a member |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(p^2 = \frac{25}{64}\) | M1 | Correct method, may be implied |
| \(p = \frac{5}{8}\) | A1 | \(p = \frac{5}{8}\) or \(P(X=20) = \frac{5}{8}\) |
| \(\text{"}\frac{5}{8}\text{"}+ q + r = 1\) or \(2qr = \frac{1}{16}\) or \(\frac{25}{64} + 2\text{"}\frac{5}{8}\text{"}q + 2\text{"}\frac{5}{8}\text{"}r + q^2 + \frac{1}{16} + r^2 = 1\) | B1 | One equation in \(q\) and \(r\) from use of \(p+q+r=1\), \(P(M=60)\) or \(\sum P(M=m)=1\) (allow ft on their value of \(p\)) |
| Any two equations from above | B1 | Two correct equations in \(q\) and \(r\) |
| \(\frac{3}{8}q - q^2 = \frac{1}{32}\) | dM1 | (dep on 1st B1) Correct method to solve simultaneous equation leading to a probability for \(q\) or \(r\) |
| \(q = \frac{1}{4}\) | A1 | Correct probability for \(q\) (dependent on all previous marks in part (d)) |
| \(P(M=50) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\) * | A1cso* | Correct solution with use of \(P(M=50) = q^2\) and all previous marks awarded |
# Question 6:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Taking a random sample is quicker/cheaper/easier (compared to asking all of the youth club members) | B1 | Any one of the given reasons. Ignore extraneous non-contradictory reasons |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| A list/register/database of all the youth club members | B1 | Idea of list(oe). Need all (oe) (e.g. complete list) and members |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The members | B1 | The members/a member |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $p^2 = \frac{25}{64}$ | M1 | Correct method, may be implied |
| $p = \frac{5}{8}$ | A1 | $p = \frac{5}{8}$ or $P(X=20) = \frac{5}{8}$ |
| $\text{"}\frac{5}{8}\text{"}+ q + r = 1$ or $2qr = \frac{1}{16}$ or $\frac{25}{64} + 2\text{"}\frac{5}{8}\text{"}q + 2\text{"}\frac{5}{8}\text{"}r + q^2 + \frac{1}{16} + r^2 = 1$ | B1 | One equation in $q$ and $r$ from use of $p+q+r=1$, $P(M=60)$ or $\sum P(M=m)=1$ (allow ft on their value of $p$) |
| Any two equations from above | B1 | Two correct equations in $q$ and $r$ |
| $\frac{3}{8}q - q^2 = \frac{1}{32}$ | dM1 | (dep on 1st B1) Correct method to solve simultaneous equation leading to a probability for $q$ or $r$ |
| $q = \frac{1}{4}$ | A1 | Correct probability for $q$ (dependent on all previous marks in part (d)) |
| $P(M=50) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ * | A1cso* | Correct solution with use of $P(M=50) = q^2$ and all previous marks awarded |6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.
\begin{enumerate}[label=(\alph*)]
\item Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
\item Suggest a suitable sampling frame.
\item Identify the sampling units.
The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70\\
When a ball is selected at random, the random variable $X$ represents the number on the ball where
$$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$
A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag.
The random variable $M$ is the mean of the 2 numbers recorded.
Given that
$$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
\item show that $\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }$
\begin{center}
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VIHV SIHII NI I IIIM I ON OC & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
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\end{center}
\includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_111_65_2525_1880}\\
\includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-24_140_233_2625_1733}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2021 Q6 [10]}}