(e) [4 marks] Suppose that daily rainfall between 10 and 20 millimetres are classified

as "moderate rain" according to a new rainfall rating system. Using the formula from

lectures, compute a 95% confidence interval for the difference in the proportions of

raining days classified as having "moderate rain" from Station 1 and from Station 2.

While you may use a function in R to verify your answer, it is expected that you show

the steps of working out the limits of the confidence interval via hand working. You

may use R to perform the calculations required for these steps. Hint: You'll have to

determine the number of daily rainfall records in the data set for each Station that

are between 10 and 20 millimetres.

(f) [4 marks] One measure of the performance of an estimator is the mean squared

error (MSE), which combines the bias and the variance of an estimator into one

measure. Suppose that X~ B(₁, p₁) and Y~ B(n2, p2) are independent binomial

random variables. Therefore the means of the random variables are E(X) = n₁p₁ and

E(Y) = n₂p2 and the variances are Var(X) = n₁p₁(1-p₁) and Var(Y) = n2p2(1-P2).

We can estimate the difference in the unknown population proportions p₁ - p2 with

the difference in the sample proportions p1-p2.

Show that the MSE of p1 - p2 is: