| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Identify distribution and parameters |
| Difficulty | Moderate -0.3 This is a straightforward S2 binomial distribution question requiring identification of parameters (n=20, p=0.4), direct probability calculation using tables/calculator, and standard transformations of binomial variables. Part (d) involves applying Var(aX+b) = a²Var(X), which is routine A-level statistics. All parts follow standard textbook patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\sim B(20, 0.4)\) | B1 | B(inomial), 20 and 0.4 all required |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(4\leq X<9)=P(X\leq 8)-P(X\leq 3)=0.5796\) | M1 A1 | For writing or using \(P(X\leq 8)-P(X\leq 3)\); awrt 0.58(0) |
| Answer | Marks | Guidance |
|---|---|---|
| \(7X-3(20-X)>0\); \(X>6\); \(1-P(X\leq 6)=1-0.2500=0.7500\) | M1, A1, M1 A1 | Using \(7X-3(20-X)\) and comparing with 0; \(X>6\); \(1-P(X\leq\text{'6'})\); awrt 0.75(0) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var}(X)=20\times0.4\times0.6\ [=4.8]\); \(\text{Var}(7X-3(20-X))=\text{Var}(10X-60)\); \(10^2\text{Var}(X)=480\) | M1, M1 A1 | Use of \(\text{Var}(X)=np(1-p)\); use of \(10^2\text{Var}(X)\) |
# Question 3:
## Part (a)
| $X\sim B(20, 0.4)$ | B1 | B(inomial), 20 and 0.4 all required |
## Part (b)
| $P(4\leq X<9)=P(X\leq 8)-P(X\leq 3)=0.5796$ | M1 A1 | For writing or using $P(X\leq 8)-P(X\leq 3)$; awrt **0.58(0)** |
## Part (c)
| $7X-3(20-X)>0$; $X>6$; $1-P(X\leq 6)=1-0.2500=0.7500$ | M1, A1, M1 A1 | Using $7X-3(20-X)$ and comparing with 0; $X>6$; $1-P(X\leq\text{'6'})$; awrt **0.75(0)** |
## Part (d)
| $\text{Var}(X)=20\times0.4\times0.6\ [=4.8]$; $\text{Var}(7X-3(20-X))=\text{Var}(10X-60)$; $10^2\text{Var}(X)=480$ | M1, M1 A1 | Use of $\text{Var}(X)=np(1-p)$; use of $10^2\text{Var}(X)$ |
---\begin{enumerate}
\item A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by $40 \%$ of the students.
\end{enumerate}
In a random sample of 20 students who sat the examination, $X$ denotes the number of students who answered the first question correctly.\\
(a) Write down the distribution of the random variable $X$\\
(b) Find $\mathrm { P } ( 4 \leqslant X < 9 )$
Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly.\\
(c) Find the probability that the total number of points scored on the first question by the 20 students is more than 0\\
(d) Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.
\hfill \mbox{\textit{Edexcel S2 2016 Q3 [10]}}