MTH6991/MTH791U/MTH791P Computational Statistics with R Exercise Sheet 3 Spring 2023

Problems for handing in

1.  (50 marks)

This question uses a dataset on QMPlus, which is not the same as the dataset for the first two exercise sheets. For each student, there should be a file called “exercise3 XYZ.txt”, where XYZ is your ID number (you need to be logged in to QMPlus).  If you cannot see a file, please send me an email.

Hand in: an R script with all the R code used, plus a separate file with your solutions - write the solutions (briefly) in your own words rather than copying and pasting any console output from R.

The dataset contains two columns, labelled“before” and “after”. Assume these are the test marks for each student in a class before and after some extra teaching sessions. We want to find out if there is a difference between the marks for each student, without assuming the data are normally distributed.

(a) Which type of data is this?  Motivate your answer, also explaining the main dif-

ferences between (two-sample) independent data and (one-sample) matched pair data.

(b) For this part, do the calculations by hand (and calculator).  Using only the first

three rows of the dataset, calculate the test statistic for the permutation test, which is the sum of the differences between the marks.  Then find the null distribution for the permutation test of whether the before and after marks have the same distribution.  Finally, calculate the p-value considering the alternative hypothesis of having different distributions.

(c) For this part, use the whole dataset.  Use wilcox .test in R to test at the 5% significance level whether there is a difference in the before and after marks. Note that there are almost certainly tied values in the full dataset, so you will need to use a normal approximation.