A researcher wants to investigate the attitudes to vaccination in England. The researcher decides to use simple random sampling to select 1000 people to take part in their investigation.

(a) Why is the sample not guaranteed to be a representative sample with respect to the age ranges and regions of England? Questions that are to be asked when collecting data include:

Q1. Have you received at least one vaccination? Options are: Yes/No.

Q2. How many vaccination doses have you received? Options are: 0, 1, 2, 3, 4.

Q3. Are you likely to take up your next offer of vaccination? Options are: 'very likely', 'fairly likely', 'not likely', 'definitely not'.

(b) For Q3, give a reason why this variable might be considered to be subjective data.

(c) For Q3, give a reason why this variable might be considered to be ordinal data.

Suppose that a pharmaceutical company has come up with a new vaccine and wishes to investigate whether this new vaccine is more effective than the best existing vaccine against COVID-19.

(d) Give a reason why a crossover design might not be suitable for this investigation.

Suppose a matched-pairs design is used and 1000 pairs are selected from the general public in England. One member of each pair is randomly selected to receive the new vaccine and the other member receives the existing vaccine. There is a measure of effectiveness for each vaccine, ranging from 0 to 100.

(e) If pairs are selected from the same region, in proportion to the population of the region as given in the file COVID-19 infections aggregate.mwx, how many pairs should be selected from Yorkshire and Humberside?

(f) State the hypotheses that can be used in this situation for performing a matched pair t-test, using notation that you should define.

(g) Give a reason why this particular hypothesis test might not turn out to be suitable.

(h) Suppose that further analysis later suggested that a Type 2 error had been made in the initial analysis. What does this mean in terms of the evidence against the null hypothesis that was observed in the initial analysis? What impact might this error have had on the choice of vaccine