[6] 1. Prove that
[P(R|D)/P(Rc|D)]/[P(R|Dc)/P(Rc|Dc)] = [P(D|R)/P(D|Rc)]/[P(Dc|R)/P(Dc|Rc)]
where Rc denotes the complement of the event R, and explain why this result is important in the interpretation of medical studies.
[4] 2. In a retrospective study, 35 men who had died from cardiovascular disease were matched with 25 men who had died from other causes. Of the 35 cases, 5 had been on a high-salt diet before they died, compared to just 2 of the control subjects. Interpret these results in terms of odds ratios and risk ratios.
[7] 3. If 7 balls are drawn at random from an urn containing 35 red balls and 25 white balls, what is the probability of getting 5 or more red balls? Can you apply this calculation to the cardiovascular study in Question 2?
[8] 4. The following table gives the mean and standard deviation of time (in sec) for the runners in a 4 x 100 m relay team. If their times are independent of each other, what is the probability that their total time will be less than 42 sec? What is the probability that they will beat 42 sec in at least 1 race out of 10? State any assumptions you make. (Note: F(2.02) = 0.9783.)
|
Runner |
Mean |
sd |
|
1 |
11.6 |
0.55 |
|
2 |
10.8 |
0.31 |
|
3 |
11.0 |
0.25 |
|
4 |
10.1 |
0.30 |
[5] 5. The following data give fork length and weight for a sample of fish. Plot a scatter diagram and compute the correlation coefficient.
|
Fish: |
1 |
2 |
3 |
4 |
5 |
|
Fork Length (cm): |
21 |
32 |
35 |
68 |
83 |
|
Weight (kg): |
0.8 |
1.8 |
1.5 |
6.9 |
14.3 |