| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Normal distribution probability then binomial/normal approximation on sample |
| Difficulty | Standard +0.3 This is a two-part question combining normal distribution percentage points with binomial-to-normal approximation. Part (a) requires finding a z-score and working backwards (routine but multi-step). Part (b) applies standard normal approximation to binomial with continuity correction. Both parts are textbook applications with no novel insight required, making it slightly easier than average. |
| Spec | 2.05g Hypothesis test using Pearson's r |
\begin{enumerate}
\item A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, $D \mathrm { ml }$, follows a normal distribution with mean 25 ml
\end{enumerate}
Given that $15 \%$ of bottles contain less than 24.63 ml\\
(a) find, to 2 decimal places, the value of $k$ such that $\mathrm { P } ( 24.63 < D < k ) = 0.45$
A random sample of 200 bottles is taken.\\
(b) Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and $k \mathrm { ml }$\\
\hfill \mbox{\textit{SPS SPS SM Statistics 2021 Q4 [8]}}