3 A researcher read a magazine article which stated that boys aged 1 to 3 prefer green to orange. It claimed that, when offered a green cube and an orange cube to play with, a boy is more likely to choose the green one. The researcher disagrees with this claim. She believes that boys of this age are equally likely to choose either colour. In order to test her belief, the researcher carried out a hypothesis test at the 5\% significance level. She offered a green cube and an orange cube to each of 10 randomly chosen boys aged 1 to 3 , and recorded the number, \(X\), of boys who chose the green cube. Out of the 10 boys, 8 boys chose the green cube.

1. 1. Assuming that the researcher's belief that either colour cube is equally likely to be chosen is valid, a student correctly calculates that \(\mathrm { P } ( X = 8 ) = 0.0439\), correct to 3 significant figures. He says that, because this value is less than 0.05 , the null hypothesis should be rejected. Explain why this statement is incorrect.
   2. Carry out the test on the researcher's claim that either colour cube is equally likely to be chosen.
2. Another researcher claims that a Type I error was made in carrying out the test. Explain why this cannot be true.
   A similar test, at the \(5 \%\) significance level, was carried out later using 10 other randomly chosen boys aged 1 to 3 .
3. Find the probability of a Type I error.