| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Moderate -0.8 This is a straightforward application of standard binomial and normal approximation techniques. Part (a) requires direct binomial calculation with n=12, part (b) is a routine normal approximation with continuity correction for n=150, and part (c) asks for the standard np>5 and nq>5 justification. All steps are textbook procedures with no problem-solving insight required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| \([= 1 - (0.063852 + 0.017414 + 0.0021768)]\) | M1 | One term: \({}^{12}C_x\ p^x\ (1-p)^{12-x}\) for \(0 < x < 12\), any \(p\) allowed |
| Correct unsimplified expression, or better | A1 | |
| \([1 - 0.083443] = 0.917\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| \([= 0.000016777 + 0.00030199 + 0.0024914 + 0.012457 + 0.042043 + 0.10090 + 0.17658 + 0.22703 + 0.21284 + 0.14189]\) | M1 | One term: \({}^{12}C_x\ p^x\ (1-p)^{12-x}\) for \(0 < x < 12\), any \(p\) allowed |
| Correct unsimplified expression with at least the first two and last terms | A1 | |
| \(0.917\) | A1 | WWW, AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Mean} = 0.6 \times 150 = 90\); \(\text{Variance} = 0.6 \times 150 \times 0.4 = 36\) | B1 | Correct mean and variance. Accept evaluated or unsimplified |
| \(P(X < 81) = P\left(Z < \frac{80.5 - 90}{6}\right)\) | M1 | Substituting *their* mean and variance into ±standardisation formula (with a numerical value for 80.5), allow \(\sigma^2\), \(\sqrt{\sigma}\), but not \(\mu \pm 0.5\) |
| M1 | Using continuity correction 80.5 or 81.5 | |
| \(\Phi(-1.5833) = 1 - 0.9433\) | M1 | Appropriate area \(\Phi\), from final process, must be probability |
| \(0.0567\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(np = 90\), \(nq = 60\) both greater than 5 | B1 | At least \(nq\) evaluated and statement \(> 5\) required |
## Question 6(a):
$1 - P(10, 11, 12) = 1 - ({}^{12}C_{10}\ 0.6^{10}\ 0.4^2 + {}^{12}C_{11}\ 0.6^{11}\ 0.4^1 + {}^{12}C_{12}\ 0.6^{12}\ 0.4^0)$
$[= 1 - (0.063852 + 0.017414 + 0.0021768)]$ | M1 | One term: ${}^{12}C_x\ p^x\ (1-p)^{12-x}$ for $0 < x < 12$, any $p$ allowed
Correct unsimplified expression, or better | A1 |
$[1 - 0.083443] = 0.917$ | A1 | AWRT
**Alternative Method:**
$P(0,1,2,\ldots,9) = {}^{12}C_0\ 0.6^0\ 0.4^{12} + {}^{12}C_1\ 0.6^1\ 0.4^{11} + \ldots + {}^{12}C_9\ 0.6^9\ 0.4^3$
$[= 0.000016777 + 0.00030199 + 0.0024914 + 0.012457 + 0.042043 + 0.10090 + 0.17658 + 0.22703 + 0.21284 + 0.14189]$ | M1 | One term: ${}^{12}C_x\ p^x\ (1-p)^{12-x}$ for $0 < x < 12$, any $p$ allowed
Correct unsimplified expression with at least the first two and last terms | A1 |
$0.917$ | A1 | WWW, AWRT
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Mean} = 0.6 \times 150 = 90$; $\text{Variance} = 0.6 \times 150 \times 0.4 = 36$ | B1 | Correct mean and variance. Accept evaluated or unsimplified |
| $P(X < 81) = P\left(Z < \frac{80.5 - 90}{6}\right)$ | M1 | Substituting *their* mean and variance into ±standardisation formula (with a numerical value for 80.5), allow $\sigma^2$, $\sqrt{\sigma}$, but not $\mu \pm 0.5$ |
| | M1 | Using continuity correction 80.5 or 81.5 |
| $\Phi(-1.5833) = 1 - 0.9433$ | M1 | Appropriate area $\Phi$, from final process, must be probability |
| $0.0567$ | A1 | AWRT |
**Total: 5 marks**
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $np = 90$, $nq = 60$ both greater than 5 | B1 | At least $nq$ evaluated and statement $> 5$ required |
**Total: 1 mark**
---6 In Questa, 60\% of the adults travel to work by car.
\begin{enumerate}[label=(\alph*)]
\item A random sample of 12 adults from Questa is taken.
Find the probability that the number who travel to work by car is less than 10 .
\item A random sample of 150 adults from Questa is taken.
Use an approximation to find the probability that the number who travel to work by car is less than 81 .
\item Justify the use of your approximation in part (b).
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q6 [9]}}