6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{740f7555-3a9a-4526-9048-39908aa8f8dd-10_684_694_239_625}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii \(4 \mathrm {~cm} , 7 \mathrm {~cm} , 9 \mathrm {~cm}\) and 10 cm . The regions are coloured as labelled in Figure 1.
A random sample of 100 children each choose a point on the target and their results are summarised in the table below.
(b) Find the value of \(r\) and the value of \(s\).
Henry obtained a test statistic of 6.188 and no groups were pooled.
(c) State what conclusion Henry should make about his claim.
Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table.
\section*{Observed frequencies}
| Colour of region | Green | Red | Blue | Yellow | Total |
| Boys | 10 | 12 | 10 | 3 | 35 |
| Girls | 12 | 27 | 15 | 11 | 65 |
(d) State suitable hypotheses to test Phoebe's belief.
Phoebe calculated the following expected frequencies to carry out a suitable test.
\section*{Expected frequencies}
| Colour of region | Green | Red | Blue | Yellow |
| Boys | 7.7 | 13.65 | 8.75 | 4.9 |
| Girls | 14.3 | 25.35 | 16.25 | 9.1 |
(e) Show how the value of 25.35 was obtained.
Phoebe carried out the test using 2 degrees of freedom and a \(10 \%\) level of significance. She obtained a test statistic of 1.411
(f) Explain clearly why Phoebe used 2 degrees of freedom.
(g) Stating your critical value clearly, determine whether or not these data support Phoebe's belief.