4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects \(2 p\) coins that are less than 6.12 grams or greater than 8.12 grams.
- The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Weight of UK two pence coins}
\includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
\end{figure}
Explain why the weight of 2 p coins can be modelled using a normal distribution. - Assume the distribution of the weight of \(2 p\) coins is normally distributed. Calculate the proportion of \(2 p\) coins that are rejected by this brand of coin counting machine.
- A manager suspects that a large batch of \(2 p\) coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.
| Summary statistics |
| Mean | | Minimum | | Median | | Maximum |
| 6.89 | 0.296 | 6.45 | 6.63 | 6.88 | 7.08 | 7.48 |
ii) Assuming the population standard deviation is 0.357 grams, test at the \(1 \%\) significance level whether the mean weight of the \(2 p\) coins in this batch is less than 7.12 grams.