Each July Rita records the number \(X\) of days on which she sees a kind of bird and models \(X\sim B(31,p)\). Earlier data suggest \(p=0.3\). In July 2022 she suspected a decrease and saw the bird on 4 days.

(a) Use the binomial distribution to test her suspicion at the 5% significance level. [5]

In July 2023 she repeats the test with the same hypotheses and significance level.

(b) Calculate the probability of a Type I error. [2]

For a different rare bird, the yearly number of sightings \(Y\) is modelled by \(B(365,0.01)\).

(c)(i) Use a suitable approximation to find \(P(Y=4)\). [3]

(c)(ii) Justify the approximation in this context. [1]