In a quiz, students have to choose the correct answer to each question from three possible options. There is only one correct answer for each question. Ethan got answers correct, and he claims that he merely guessed the answer to every question but his teacher believes he used some knowledge in the quiz. She uses the null hypothesis to test her belief at the 10% significance level.
(i)
If the teacher wishes to test to see if Ethan was trying to get the answers correct, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.
(ii)
If the teacher wishes to test to see if Ethan was trying to get the answers incorrect, rather than guessing them at random, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.
(iii)
If the teacher wishes to test to see whether Ethan was not guessing the answers at random, but she is uncertain whether he was using his knowledge to get them right or to get them wrong, write down the alternative hypothesis she should use and explain the conditions under which the null hypothesis would be rejected.