| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Small sample binomial probability |
| Difficulty | Moderate -0.3 This is a straightforward application of standard binomial and Poisson distribution calculations using tables or calculators. Part (a) requires a single binomial probability calculation with clearly stated parameters. Parts (b)(i-iii) involve direct Poisson probability lookups and recognizing that the sum of independent Poisson variables is Poisson. All parts are routine bookwork with no problem-solving insight required, making it slightly easier than average. |
| Spec | 5.02c Linear coding: effects on mean and variance5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| \(X\) = no. with blood disorder, \(X \sim B(25, 0.7)\), \(P(X > 15) = P(X \geq 16)\). Consider \(X' \sim B(25, 0.3)\): \(P(X \geq 16) = P(X' \leq 9) = 0.8106\) | B3,2,1 | B3 \(0.81 \leq p \leq 0.811\); B2 for \(0.902 \leq p \leq 0.9022\); B1 for \(0.5 \leq p \leq 0.95\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim P_0(2.6)\), \(P(X \leq 5) = 0.951\) | B1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y \sim P_0(4.9)\), \(P(Y=10) = \frac{e^{-4.9} \times (4.9)^{10}}{10!} = 0.0164\) | B1, M1, A1 | \(\lambda = 4.9\) stated or used; AWFW 0.016 to 0.0165 |
| Answer | Marks | Guidance |
|---|---|---|
| \(T \sim P_0(7.5)\), \(P(T > 16) = 1 - P(T \leq 16) = 1 - 0.9980 = 0.002\) | B1ft, M1, A1 | \(2.6 +\) (their mean in (ii)); for 0.9980; CAO (0.00196) |
## Question 5(a):
| $X$ = no. with blood disorder, $X \sim B(25, 0.7)$, $P(X > 15) = P(X \geq 16)$. Consider $X' \sim B(25, 0.3)$: $P(X \geq 16) = P(X' \leq 9) = 0.8106$ | B3,2,1 | **B3** $0.81 \leq p \leq 0.811$; **B2** for $0.902 \leq p \leq 0.9022$; **B1** for $0.5 \leq p \leq 0.95$ |
## Question 5(b)(i):
| $X \sim P_0(2.6)$, $P(X \leq 5) = 0.951$ | B1 | AWRT |
## Question 5(b)(ii):
| $Y \sim P_0(4.9)$, $P(Y=10) = \frac{e^{-4.9} \times (4.9)^{10}}{10!} = 0.0164$ | B1, M1, A1 | $\lambda = 4.9$ stated or used; AWFW 0.016 to 0.0165 |
## Question 5(b)(iii):
| $T \sim P_0(7.5)$, $P(T > 16) = 1 - P(T \leq 16) = 1 - 0.9980 = 0.002$ | B1ft, M1, A1 | $2.6 +$ (their mean in (ii)); for 0.9980; CAO (0.00196) |
---5
\begin{enumerate}[label=(\alph*)]
\item In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random.
Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
\item \begin{enumerate}[label=(\roman*)]
\item In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people.
A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
\item In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people.
A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
\item The random variable $T$ denotes the total number of people in the two selected towns who have this blood disorder.
Write down the distribution of $T$ and hence determine $\mathrm { P } ( T > 16 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2010 Q5 [10]}}