- John thinks that a person's eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in Table 1.
\begin{table}[h]
| \multirow{2}{*}{} | Hair colour |
| | Black | Brown | Red | Blonde | Total |
| \multirow{5}{*}{Eye colour} | Brown | 45 | 125 | 15 | 58 | 243 |
| Blue | 34 | 90 | 10 | 58 | 192 |
| Hazel | 20 | 38 | 16 | 26 | 100 |
| Green | 6 | 29 | 7 | 23 | 65 |
| Total | 105 | 282 | 48 | 165 | 600 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
John carries out a \(\chi ^ { 2 }\) test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2.
\begin{table}[h]
| \multirow{2}{*}{} | Hair colour |
| | Black | Brown | Red | Blonde |
| \multirow{4}{*}{Eye colour} | Brown | 42.5 | 114.2 | 19.4 | 66.8 |
| Blue | 33.6 | 90.2 | 15.4 | 52.8 |
| Hazel | 17.5 | 47 | 8 | 27.5 |
| Green | 11.4 | 30.6 | 5.2 | 17.9 |
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{table}
- Show how the value 47 in Table 2 has been calculated.
- Write down the number of degrees of freedom John should use in this \(\chi ^ { 2 }\) test.
Given that the value of the \(\chi ^ { 2 }\) statistic is 20.6 , to 3 significant figures,
- find the smallest value of \(\alpha\) for which the null hypothesis will be rejected at the \(\alpha \%\) level of significance.
- Use the data from Table 1 to test at the \(5 \%\) level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.