| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find minimum/maximum n for probability condition |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard binomial distribution calculations (part a) and normal approximation with continuity correction (part b). Part (a) requires direct use of binomial probability formula, while part (b) involves routine application of normal approximation and inverse normal tables to find n. The mechanics are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(X\sim B(10, 0.6)\) or \(Y\sim B(10, 0.4)\) written in either part (i) or (ii) | B1 | |
| \(P(X=6)=(0.6)^6(0.4)^4\dfrac{10!}{6!4!}\) or \(P(Y=4)=(0.4)^4(0.6)^6\dfrac{10!}{6!4!}\) | M1 | Allow \(^{10}C_6\) oe; NB use of Poisson gains M0A0 |
| \(= 0.2508\) | A1 | awrt \(0.251\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(P(X<9)=1-(P(X=10)+P(X=9))\) or \(P(Y<9)=1-P(Y\leq 1)\) | M1 | Writing or using \(1-(P(X=10)+P(X=9))\) if \(B(10,0.6)\); or \(1-P(Y\leq 1)\) if \(B(10,0.4)\); NB Poisson gives M0A0 |
| \(=1-(0.6)^{10}-(0.6)^9(0.4)^1\dfrac{10!}{9!1!}\) or \(=1-0.0464\) | ||
| \(= 0.9536\) | A1 | awrt \(0.954\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(X\sim B(50, 0.6)\) or \(Y\sim B(50, 0.4)\) | M1 | 1st M1 for writing or using either \(B(50,0.6)\) or \(B(50,0.4)\) |
| \(P(X | M1 | 2nd M1: \(P(Y>50-n)\geq 0.9\) or \(P(Y\leq 50-n)\leq 0.1\) or \(P(X<34)=0.8439\) awrt \(0.844\) or \(P(X<35)=0.9045\) awrt \(0.904/0.905\); or \(50-n\leq 15\) or \(50-n\leq 16\); allow different letters |
| \(50-n\leq 15\), so \(n\geq 35\), thus \(n=35\) | A1 | cao \(35\). Do not accept \(n\geq 35\) for final A1 |
| SC: use of normal \(N(30,12)\) leading to answer of \(35\) | M1 M0 A0 | Special case |
# Question 8:
## Part (a)(i)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $X\sim B(10, 0.6)$ or $Y\sim B(10, 0.4)$ written in either part (i) or (ii) | B1 | |
| $P(X=6)=(0.6)^6(0.4)^4\dfrac{10!}{6!4!}$ or $P(Y=4)=(0.4)^4(0.6)^6\dfrac{10!}{6!4!}$ | M1 | Allow $^{10}C_6$ oe; NB use of Poisson gains M0A0 |
| $= 0.2508$ | A1 | awrt $0.251$ |
## Part (a)(ii)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $P(X<9)=1-(P(X=10)+P(X=9))$ or $P(Y<9)=1-P(Y\leq 1)$ | M1 | Writing or using $1-(P(X=10)+P(X=9))$ if $B(10,0.6)$; or $1-P(Y\leq 1)$ if $B(10,0.4)$; NB Poisson gives M0A0 |
| $=1-(0.6)^{10}-(0.6)^9(0.4)^1\dfrac{10!}{9!1!}$ or $=1-0.0464$ | | |
| $= 0.9536$ | A1 | awrt $0.954$ |
## Part (b)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $X\sim B(50, 0.6)$ or $Y\sim B(50, 0.4)$ | M1 | 1st M1 for writing or using either $B(50,0.6)$ or $B(50,0.4)$ |
| $P(X<n)\geq 0.9$, so $P(Y>50-n)\geq 0.9$ or $P(Y\leq 50-n)\leq 0.1$ | M1 | 2nd M1: $P(Y>50-n)\geq 0.9$ or $P(Y\leq 50-n)\leq 0.1$ or $P(X<34)=0.8439$ awrt $0.844$ or $P(X<35)=0.9045$ awrt $0.904/0.905$; or $50-n\leq 15$ or $50-n\leq 16$; allow different letters |
| $50-n\leq 15$, so $n\geq 35$, thus $n=35$ | A1 | cao $35$. Do not accept $n\geq 35$ for final A1 |
| SC: use of normal $N(30,12)$ leading to answer of $35$ | M1 M0 A0 | Special case |\begin{enumerate}
\item In a large restaurant an average of 3 out of every 5 customers ask for water with their meal.
\end{enumerate}
A random sample of 10 customers is selected.\\
(a) Find the probability that\\
(i) exactly 6 ask for water with their meal,\\
(ii) less than 9 ask for water with their meal.
A second random sample of 50 customers is selected.\\
(b) Find the smallest value of $n$ such that
$$\mathrm { P } ( X < n ) \geqslant 0.9$$
where the random variable $X$ represents the number of these customers who ask for water.\\
\hfill \mbox{\textit{Edexcel S2 2012 Q8 [8]}}