| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of mean or linear combination |
| Difficulty | Standard +0.3 This is a straightforward S2 sampling distribution question requiring listing outcomes, using given probabilities to find remaining probabilities (simple algebra with ratios), and applying binomial probability P(Y≥1)=1-P(Y=0) with logarithms. All techniques are standard for S2 with no novel insight required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| 44, 46, 48, 66, 68, 88 (NB: 64 same as 46; 84 same as 48; 86 same as 68) | B1B1 | B1: at least 4 different pairs (ignore incorrect extras); B1: 6 different pairs with no incorrect extras |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\bar{x}\): 4, 5, 6, 7, 8 with probabilities \(\frac{1}{4}\), \(\frac{3}{10}\), \(\frac{29}{100}\), \(\frac{3}{25}\), \(\frac{1}{25}\) | B1B1M1M1A1 | B1: values 4,5,6,7,8 only, no extras/omissions; B1: \(P(X=4)=\frac{1}{2}\)... wait — \(P(\bar{X}=4)=\frac{1}{4}\), \(P(\bar{X}=6)=\frac{3}{10}\), \(P(\bar{X}=8)=\frac{1}{5}\); M1: correct method for one of \(P(5)\), \(P(6)\), \(P(7)\); M1: correct method for two of \(P(5)\), \(P(6)\), \(P(7)\); A1: fully correct table |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - \left(\frac{24}{25}\right)^n > 0.9\) or \(\left(\frac{24}{25}\right)^n < 0.1\) oe | M1 | May use \(=\) or \(\leq\) instead of \(<\), \(=\) or \(\geq\) instead of \(>\); do not award \(\left(\frac{24}{25}\right)^n > 0.1\) oe |
| \(n > 56.4\) | A1 | Ignore any \(n>\), \(n<\), \(n=\) etc.; awrt 56.4; may be implied by \(n = 57\) |
| \(n = 57\) | A1 | cao; do not allow \(n > 57\) or \(n < 57\); check no incorrect working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Values for \(n = 50\) to \(60\) with correct probabilities | M1 | At least 2 trials for \(50 \leq n \leq 60\) shown with correct probabilities |
| Trial for \(n = 56\) and \(n = 57\) shown with correct probabilities | A1 | |
| \(n = 57\) | A1 | cao; do not allow \(n > 57\) or \(n < 57\); do not award if alternative values given |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| 44, 46, 48, 66, 68, 88 (NB: 64 same as 46; 84 same as 48; 86 same as 68) | B1B1 | B1: at least 4 different pairs (ignore incorrect extras); B1: 6 different pairs with no incorrect extras |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\bar{x}$: 4, 5, 6, 7, 8 with probabilities $\frac{1}{4}$, $\frac{3}{10}$, $\frac{29}{100}$, $\frac{3}{25}$, $\frac{1}{25}$ | B1B1M1M1A1 | B1: values 4,5,6,7,8 only, no extras/omissions; B1: $P(X=4)=\frac{1}{2}$... wait — $P(\bar{X}=4)=\frac{1}{4}$, $P(\bar{X}=6)=\frac{3}{10}$, $P(\bar{X}=8)=\frac{1}{5}$; M1: correct method for one of $P(5)$, $P(6)$, $P(7)$; M1: correct method for two of $P(5)$, $P(6)$, $P(7)$; A1: fully correct table |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - \left(\frac{24}{25}\right)^n > 0.9$ or $\left(\frac{24}{25}\right)^n < 0.1$ oe | M1 | May use $=$ or $\leq$ instead of $<$, $=$ or $\geq$ instead of $>$; do not award $\left(\frac{24}{25}\right)^n > 0.1$ oe |
| $n > 56.4$ | A1 | Ignore any $n>$, $n<$, $n=$ etc.; awrt 56.4; may be implied by $n = 57$ |
| $n = 57$ | A1 | cao; do not allow $n > 57$ or $n < 57$; check no incorrect working |
**Alternative – trial and error:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Values for $n = 50$ to $60$ with correct probabilities | M1 | At least 2 trials for $50 \leq n \leq 60$ shown with correct probabilities |
| Trial for $n = 56$ and $n = 57$ shown with correct probabilities | A1 | |
| $n = 57$ | A1 | cao; do not allow $n > 57$ or $n < 57$; do not award if alternative values given |6. A bag contains a large number of counters with one of the numbers 4,6 or 8 written on each of them in the ratio $5 : 3 : 2$ respectively.
A random sample of 2 counters is taken from the bag.
\begin{enumerate}[label=(\alph*)]
\item List all the possible samples of size 2 that can be taken.
The random variable $M$ represents the mean value of the 2 counters.\\
Given that $\mathrm { P } ( M = 4 ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( M = 8 ) = \frac { 1 } { 25 }$
\item find the sampling distribution for $M$.
A sample of $n$ sets of 2 counters is taken. The random variable $Y$ represents the number of these $n$ sets that have a mean of 8
\item Calculate the minimum value of $n$ such that $\mathrm { P } ( Y \geqslant 1 ) > 0.9$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q6 [10]}}