Question 1. [19 marks]

Suppose that daily rainfall records (in millimetres) are observed at two weather stations.

A random sample of 50 positive daily rainfall records from Station 1 and 45 positive daily

rainfall records from Station 2 are provided in the file rainfall.txt on Canvas. From

the data, the mean daily rainfall for Station 1 is about 20.49 millimetres and the mean

daily rainfall for Station 2 is about 7.79 millimetres, as shown below:

rainfall = read.table ("rainfall.txt", header = TRUE)

station1 = rainfall$rainfall [rainfall$station == 1)

station2= rainfall$rainfall [rainfall$station == 2]

mean(station1)

## [1] 20.4936

mean (station2)

## [1] 7.787333

(a) [2 marks] Using R, produce a QQ-plot comparing the daily rainfall records to

a normal distribution using the qqnorm function for both the rainfall data from

Station 1 and the rainfall data from Station 2. What do these plots suggest about the

distribution of daily rainfall records? Include R code and the plots in your answer.

(b) [4 marks] Using the formula from lectures, compute a 95% confidence interval

for the ratio of the population variance in daily rainfall records between the selected

rainfall data from Station 1 and the selected rainfall data 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.

(c) [1 mark] Based on your confidence interval from part (b), is it reasonable to assume

the variance in daily rainfall records is the same for Station 1 and Station 2? Explain

why or why not.

(d) [4 marks] Using the formula from lectures consistent with your conclusion to part

(c), compute a 95% confidence interval for the difference in the population average

rainfall from Station 1 and rainfall 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.