| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Find p then binomial probability |
| Difficulty | Standard +0.3 This is a standard S2 question combining normal distribution, binomial probability, and Poisson approximation. Part (a) requires inverse normal calculation using percentage points (routine), part (b) is straightforward binomial with p=0.05, and part (c) applies the standard Poisson approximation technique taught in S2. All steps follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left[P(D<108)=\right] P\left(Z < \frac{108-112.4}{\sigma}\right) = 0.05\) | M1 | For standardisation using 108 (condone 107.5), 112.4 and \(\sigma\) set equal to \(z\) where \(1.5< |
| \(\Rightarrow \frac{108-112.4}{\sigma} = -1.6449\) | M1 | For correct equation awrt \(-1.6449\) (allow awrt 1.6449 if compatible with their equation) |
| \(\sigma = 2.6749...\) days (calc 2.67501...) | A1 awrt 2.67/2.68 | NB M1 M0 A1 is possible |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(J \sim B(25, 0.05)\) | ||
| \(\left[P(J\geq 4)=\right] 1 - P(J \leq 3) = 1 - 0.9659\) | M1 | For \(1-P(J \leq 3)\) or \(1-0.9659\) |
| \(= 0.0341\) (calc 0.034090...) | A1 awrt 0.0341 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(T \sim Po[200 \times "0.0341"] = 6.82\) (calc 6.8181...) | M1 | For writing or using correct Poisson model ft their part (b); may be implied by 0.00853(73) |
| \(\left[P(T\geq 2)=\right] 1-P(X \leq 1) = 1-\left(e^{-"6.82"} + e^{-"6.82"} \times "6.82"\right)\) | M1 | For writing or using \(1-\left(e^{-"\lambda"} + e^{-"\lambda"} \times "\lambda"\right)\) where \(1<\lambda<200\); allow \(1-P(X \leq 1)\) if Poisson stated or used |
| \(= 0.99146...\) (calc 0.99144...) | dA1 awrt 0.991 | Dep on both M marks; NB Binomial gives awrt 0.992; allow 0.9915 if both M marks awarded |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left[P(D<108)=\right] P\left(Z < \frac{108-112.4}{\sigma}\right) = 0.05$ | M1 | For standardisation using 108 (condone 107.5), 112.4 and $\sigma$ set equal to $z$ where $1.5<|z|<2.5$ |
| $\Rightarrow \frac{108-112.4}{\sigma} = -1.6449$ | M1 | For correct equation awrt $-1.6449$ (allow awrt 1.6449 if compatible with their equation) |
| $\sigma = 2.6749...$ days (calc 2.67501...) | A1 awrt 2.67/2.68 | NB M1 M0 A1 is possible |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $J \sim B(25, 0.05)$ | | |
| $\left[P(J\geq 4)=\right] 1 - P(J \leq 3) = 1 - 0.9659$ | M1 | For $1-P(J \leq 3)$ or $1-0.9659$ |
| $= 0.0341$ (calc 0.034090...) | A1 awrt 0.0341 | |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $T \sim Po[200 \times "0.0341"] = 6.82$ (calc 6.8181...) | M1 | For writing or using correct Poisson model ft their part (b); may be implied by 0.00853(73) |
| $\left[P(T\geq 2)=\right] 1-P(X \leq 1) = 1-\left(e^{-"6.82"} + e^{-"6.82"} \times "6.82"\right)$ | M1 | For writing or using $1-\left(e^{-"\lambda"} + e^{-"\lambda"} \times "\lambda"\right)$ where $1<\lambda<200$; allow $1-P(X \leq 1)$ if Poisson stated or used |
| $= 0.99146...$ (calc 0.99144...) | dA1 awrt 0.991 | Dep on both M marks; NB Binomial gives awrt 0.992; allow 0.9915 if both M marks awarded |
---\begin{enumerate}
\item The length of pregnancy for a randomly selected pregnant sheep is $D$ days where
\end{enumerate}
$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$
Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,\\
(a) find the value of $\sigma$
Qiang selects 25 pregnant sheep at random from a large flock.\\
(b) Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days.
Charlie takes 200 random samples of 25 pregnant sheep.\\
(c) Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.
\hfill \mbox{\textit{Edexcel S2 2024 Q2 [8]}}