| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Normal approximation to binomial |
| Difficulty | Standard +0.3 This is a straightforward binomial distribution question requiring standard calculations: part (i) uses direct binomial probability with small n, part (ii) applies normal approximation with continuity correction for large n, and part (iii) checks np>5 and nq>5. All techniques are routine for S1 level 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 probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| \[= 0.606\] | M1, M1, A1 3 | Bin term \(p^r(1-p)^{s-r} \times ^sC_r\), seen any \(p\) or Correct unsimplified answer or Correct ans |
| Answer | Marks | Guidance |
|---|---|---|
| \[P(> 114) = P\left(z > \frac{114.5 - 120}{\sqrt{30}}\right) = P(z > -1.004) = \Phi(1.004) = 0.842\] | B1, M1, M1, M1, A1 5 | Unsimplified mean and var correct or Standardising, need sq rt or Cont correction either 114.5 or 113.5 or Correct area consistent with their working |
| (iii) np and nq both \(> 5\) | B1 1 | Need both |
**(i)** $$P(4, 5, 6) = (0.75)^4(0.25)^1 \times ^4C_4 + (0.75)^3(0.25)^2 \times ^5C_5 + (0.75)^0(0.25)^2 \times ^6C_6$$
$$= 0.606$$ | M1, M1, A1 3 | Bin term $p^r(1-p)^{s-r} \times ^sC_r$, seen any $p$ or Correct unsimplified answer or Correct ans
**(ii)** $$np = 160 \times 0.75 = 120 \quad npq = 30$$
$$P(> 114) = P\left(z > \frac{114.5 - 120}{\sqrt{30}}\right) = P(z > -1.004) = \Phi(1.004) = 0.842$$ | B1, M1, M1, M1, A1 5 | Unsimplified mean and var correct or Standardising, need sq rt or Cont correction either 114.5 or 113.5 or Correct area consistent with their working
**(iii)** np and nq both $> 5$ | B1 1 | Need bothIn Marumbo, three quarters of the adults own a cell phone.
\begin{enumerate}[label=(\roman*)]
\item A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
\item A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
\item Justify the use of your approximation in part (ii). [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2014 Q7 [9]}}