| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find minimum/maximum n for probability condition |
| Difficulty | Standard +0.3 This is a straightforward application of normal approximation to binomial (part i) and Poisson (part ii). Students need to apply continuity correction, standardize to z-scores, and use tables—all standard techniques. Part (ii) requires solving backwards from a probability to find N, which adds minor complexity but remains routine for Further Maths students. Slightly above average difficulty due to the two-distribution setup and inverse normal calculation. |
| Spec | 2.04d Normal approximation to binomial2.04f Find normal probabilities: Z transformation |
**Question 5(i)**
$np = 100 \times \frac{1}{5} = 20$ and $npq = 20 \times \frac{4}{5} = 16$ **B1**
Standardisation in either **(a)** or **(b)** **M1**
**(a)** $z = \frac{14.5 - 20}{4} = -1.375 \Rightarrow P(\geq 15) = 0.915$ (AWFW [0.915, 0.916]) **A1↓A1**
(ft on variance of 20)
**(b)** $z = \frac{12.5 - 20}{4} = -1.875 \Rightarrow P(<12) = 1 - 0.9696 = 0.0304$ **A1A1**
(AWFW [0.0303, 0.0304]) [6]
**Question 5(ii)**
mean = variance = $36 \Rightarrow$ S.D. $= 6$ **B1**
$z = 1.645$ **B1**
$\frac{\left(N + \frac{1}{2}\right) - 36}{6} > 1.645$ (Allow working with equality, but must be $\left(N + \frac{1}{2}\right)$.) **M1A1**
$\Rightarrow N > 45.37 \quad \therefore$ least $N = 46$ **A1** [5] **[11]**5 (i) The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space
\begin{enumerate}[label=(\alph*)]
\item at least 15 times,
\item no more than 12 times.\\
(ii) The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than $N$ parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of $N$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2012 Q5 [11]}}