4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of \(10 \mathrm {~m} ^ { 2 }\) over a period of months.

1. Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
   Fig. 4 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D
   1	Number of coins found	Observed frequency	Expected frequency	Chi-squared contribution
   2	0	13	13.8069	0.0472
   3	1	28
   4	2	30	27.0643	0.3184
   5	3	14	17.8625	0.8352
   6	4	10	8.8419	0.1517
   7	\(\geqslant 5\)	5		0.0015

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{table}
2. Showing your calculations, find the missing values in each of the following cells.
   • C3
   • C7
   • D3
   • Explain why the numbers for 5, 6 and more than 6 coins found have been combined into the single category of at least 5 coins found, as shown in the spreadsheet.
   • Complete the hypothesis test at the \(5 \%\) level of significance.
   For the rest of this question, you should assume that the number of coins that Zara finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution with mean 1.98.
   Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per \(10 \mathrm {~m} ^ { 2 }\) area is modelled by a Poisson distribution with mean 0.42 .
3. Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of \(10 \mathrm {~m} ^ { 2 }\).
4. Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of \(100 \mathrm {~m} ^ { 2 }\).