7 It is thought that a person's eye colour is related to the reaction of the person's skin to ultra-violet light. As part of a study, a random sample of 140 people were treated with a standard dose of ultra-violet light. The degree of reaction was classified as None, Mild or Strong. The results are given in Table 1. The corresponding expected frequencies for a \(\chi ^ { 2 }\) test of association between eye colour and reaction are shown in Table 2. \begin{table}[h] \end{table} \begin{table}[h] \end{table}

1. (a) State suitable hypotheses for the test.
   (b) Show how the expected frequency of 18.11 in Table 2 is obtained.
   (c) Show that the three cells in the top row together contribute 4.53 to the calculated value of \(\chi ^ { 2 }\), correct to 2 decimal places.
   (d) You are given that the total calculated value of \(\chi ^ { 2 }\) is 12.78 , correct to 2 decimal places. Give the smallest value of \(\alpha\) obtained from the tables for which the null hypothesis would be rejected at the \(\alpha \%\) significance level.
2. Test, at the \(5 \%\) significance level, whether the proportions of people in the whole population with blue eyes, brown eyes and other colours are in the ratios \(2 : 2 : 1\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }