| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 8 |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial distribution for small n (part a) and normal approximation to binomial for large n (part b). All steps are standard textbook procedures: using binomial probability formula, calculating mean np and variance np(1-p), applying continuity correction, and using normal tables. No problem-solving insight required, just routine application of well-practiced techniques. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.04f Find normal probabilities: Z transformation |
13. Sam is playing a computer game.
When Sam earns a reward in the game, she randomly receives either a Silver reward or a Gold reward.
Each time that Sam earns a reward, the probability of receiving a Gold reward is 0.4 One day Sam plays the computer game and earns 11 rewards.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that she receives
\begin{enumerate}[label=(\roman*)]
\item exactly 2 Gold rewards,
\item at least 5 Gold rewards.
In the next month Sam earns 300 rewards.\\
She decides to use a Normal distribution to estimate the probability that she will receive at least 135 Gold rewards.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the mean and variance of this Normal distribution.
\item Estimate the probability that Sam will receive at least 135 Gold rewards.\\[0pt]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q13 [8]}}