Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number X of days on which she sees it. She models the distribution of X by B(31, p), where p is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that p = 0.3, but in 2022 Rita suspected that the value of p had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. (a)Use the binomial distribution to test at the 5% significance level whether Rita's suspicion is justified.[5]
2. (b)In July 2023, she noted the value of X and carried out another test at the 5% significance level using the same hypotheses. Calculate the probability of a Type I error.[2]
3. (c)Rita models the number of sightings, Y, per year of a different, very rare, kind of bird by the distribution B(365, 0.01).
4. (c)(i)Use a suitable approximating distribution to find P(Y = 4).[3]
5. (c)(ii)Justify your approximating distribution in this context.[1]