| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Small sample binomial probability |
| Difficulty | Standard +0.3 Part (a) is a straightforward binomial probability calculation using tables or calculator (B(20,0.4)), requiring only direct application of P(5≤X<8)=P(X≤7)-P(X≤4). Part (b) involves standard normal approximation with continuity correction: finding n where P(X<n)≈P(Z<(n-0.5-56)/√33.6)=0.0239, requiring inverse normal lookup and solving. This is a routine S2 question testing standard techniques 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim B(20, 0.4)\) | M1 | Writing or using \(B(20, 0.4)\) |
| \(P(5 \leq X < 8) = P(X \leq 7) - P(X \leq 4)\) or \(0.4159 - 0.0510\) | M1 | For writing or using \(P(X \leq 7) - P(X \leq 4)\) or \(P(X=5)+P(X=6)+P(X=7)\) |
| \(= 0.3649\) | A1 | awrt 0.365 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Y \sim N(56, 33.6)\) | M1A1 | For writing or using \(N(56,\ldots)\); for writing or using \(N(56, 33.6)\) |
| \(\dfrac{n - 0.5 - \text{"56"}}{\sqrt{\text{"33.6"}}} = -1.98\) | M1M1B1 | For standardising (allow \(\pm\)) using their "56" and "33.6"; M1 for continuity correction \(-0.5\); B1 for using \(\pm 1.98\) or better |
| \(n = 45\) | A1cao | 45 — must see a correct continuity correction or awrt 45.02 or awrt 45.03. 45 from no working is 0/6 |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim B(20, 0.4)$ | M1 | Writing or using $B(20, 0.4)$ |
| $P(5 \leq X < 8) = P(X \leq 7) - P(X \leq 4)$ or $0.4159 - 0.0510$ | M1 | For writing or using $P(X \leq 7) - P(X \leq 4)$ or $P(X=5)+P(X=6)+P(X=7)$ |
| $= 0.3649$ | A1 | awrt 0.365 |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Y \sim N(56, 33.6)$ | M1A1 | For writing or using $N(56,\ldots)$; for writing or using $N(56, 33.6)$ |
| $\dfrac{n - 0.5 - \text{"56"}}{\sqrt{\text{"33.6"}}} = -1.98$ | M1M1B1 | For standardising (allow $\pm$) using their "56" and "33.6"; M1 for continuity correction $-0.5$; B1 for using $\pm 1.98$ or better |
| $n = 45$ | A1cao | 45 — must see a correct continuity correction or awrt 45.02 or awrt 45.03. 45 from no working is 0/6 |\begin{enumerate}
\item The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, $X$, represents the number of people who complete the task in less than 15 minutes.\\
(a) Find $\mathrm { P } ( 5 \leqslant X < 8 )$
\end{enumerate}
Mia takes a random sample of 140 people.\\
Using a normal approximation, the probability that fewer than $n$ of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.\\
(b) Find the value of $n$
Show your working clearly.
\hfill \mbox{\textit{Edexcel S2 2022 Q4 [9]}}