9 Chapters 9 and 10 problems

  1. An experiment is planned to test a new treatment to help people to quit smoking. Volunteer smokers (who wish to quit) will be randomly allocated to one of two groups. In group 1, the volunteers will use the new treatment for two months. They will be recorded as “quitting” if they do not smoke at all for a six month period following treatment. The second group will act as a control group, in which the volunteers will instead be given a placebo, but then monitored in the same way. A total of \(2n\) volunteers will be recruited, with \(n\) volunteers allocated to each group.
  • The data (the proportions quitting in each group) will be analysed with a Neyman-Pearson hypothesis test, with size 0.05.
  • The investigators want to detect a difference of at least 10% in the proportions quitting in the two groups. They suggest assuming that 55% will quit in the treatment group, and 45% will quit in the control group.
  • The investigators want their study to have a power of 80%.

What sample size \(n\) per group would you recommend? Some R output to help you is below.

qnorm(c(0.8, 0.975))
## [1] 0.8416 1.9600
  1. A random sample of 993 English voters were asked to name their favoured party. Each voter was also asked if they had attended or were currently attending university. The following table shows the results of cross-tabulating university education and political preferences of the voters.
Con Lab Lib Other
University: no 226 214 45 80
University: yes 151 185 42 50

Test the hypothesis that voting preferences are the same regardless of university education. Some R output to help is as follows.

qchisq(c(0.95, 0.99, 0.999), 3)
## [1]  7.815 11.345 16.266
  1. In a survey of 237 (Statistics) students from the University of Adelaide, amongst other variables, smoking habits (recorded here as one of Never",Occasional”, Regular" orHeavy”) and exercise levels (recorded here as one of Regular'' orsome/none’’) were observed. A contingency table is given below, with smoking status in the rows, and exercise in the columns.
##            Regular Some/none
## Never           87       102
## Occasional      12         7
## Regular          9         8
## Heavy            7         4

Test the hypothesis that exercise is independent of smoking status.

Some R output to help is as follows.

qchisq(c(0.95, 0.99, 0.999), 3)
## [1]  7.815 11.345 16.266
  1. Challenge problem. A study compared two treatments A and B for kidney stones. For each treatment, the number of times the treatment was successful out of the total number of patients treated is given below.
Treatment A Treatment B
273/350 289/350
  1. Just by looking at the numbers, which treatment appears to be the most effective?

It was also recorded whether each stone was ‘small’ or ‘large’:

Treatment A Treatment B
small 81/87 234/270
large 192/263 55/80
  1. Considering small stones and large stones separately, and again, just by looking at the numbers, which treatment appears to be the most effective? Why is your conclusion different from that in part (a)?