| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Basic E(X) and Var(X) calculation |
| Difficulty | Easy -1.2 This is a straightforward application of basic binomial distribution formulas E(X)=np and Var(X)=np(1-p). Part (a) requires simple algebraic manipulation (5=0.04n), and part (b) requires direct substitution into the standard deviation formula. Both parts are routine recall with minimal problem-solving, making this easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(n, 0.04)\) | B1 | Implied |
| \(E(X) = np\) | M1 | Use of \(np = 5\) |
| \(5 = 0.04n\) | A1 | |
| \(n = 125\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 3\), \(np = 3\) | B1 | |
| \(\text{sd} = \sqrt{npq} = \sqrt{3(1-0.04)}\) | M1 | Use of \(npq\) |
| \(= \sqrt{2.88} = 1.70\) | A1, A1 | awrt 1.70 |
**1(a)**
| $X \sim B(n, 0.04)$ | B1 | Implied |
| $E(X) = np$ | M1 | Use of $np = 5$ |
| $5 = 0.04n$ | A1 | |
| $n = 125$ | A1 | |
**1(b)**
| $E(X) = 3$, $np = 3$ | B1 | |
| $\text{sd} = \sqrt{npq} = \sqrt{3(1-0.04)}$ | M1 | Use of $npq$ |
| $= \sqrt{2.88} = 1.70$ | A1, A1 | awrt 1.70 |
---\begin{enumerate}
\item It is estimated that $4 \%$ of people have green eyes. In a random sample of size $n$, the expected number of people with green eyes is 5 .\\
(a) Calculate the value of $n$.
\end{enumerate}
The expected number of people with green eyes in a second random sample is 3 .\\
(b) Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .\\
(a) Calculate the value of $n$ -
The expected number of people with green eyes in a second random sample is 3 .\\
(b) sample. C) T. " D\\
\hfill \mbox{\textit{Edexcel S2 2005 Q1 [7]}}