Question 1
At a particular hospital, 98% of all babies survive delivery. However, 15% of all births involve Caesarean sections, and when a Caesarean section is performed the baby survives 96% of the time.
A pregnant woman is chosen at random from the hospital. She does not have a Caesarean section. What is the probability that her baby survives delivery? Support your answer with a probability tree.
Question 2
The number of patients arriving at a hospital's accident and emergency unit during the peak period on a Saturday night and who require immediate attention has been shown to follow a Poisson distribution with parameter p = 3. The unit can handle up to four such patients during this period. If more than four patients arrive in this period, the patients in excess of four must be sent to another hospital. The following questions refer to the peak period on Saturdays.
(a) On a given Saturday, what is the probability of having to send patients to another hospital?
(b) By how much must present facilities be increased to permit handling of all patients arriving on at least 95% of Saturdays?
(c) What is the expected number of patients arriving each Saturday?
(d) What is the most probable number of patients arriving each Saturday?
(e) Give the main assumptions we make when we model the above situation using a Poisson distribution.
Question 3
In a small clinical trial. 10 patients with chronic pain were randomly assigned to receive either drug A or drug B for one month. After completing the month, they then had a two-week "wash-out" period during which no drug was taken, and then received the alternate drug for one month. At the end of the study, they were asked to state which drug they preferred because it gave them greater relief from the pain Each patient had to choose one drug. Eight patients said that they preferred drug A.
(a) If drug A and drug B were truly equal in their effects in reducing pain. what would be the probability that a patient expressed a preference for drug A over drug B.
(b) How many ways are there of obtaining 10 patients expressing a preference for drug A. Repeat for 9 patients. 8 patients. ... 1, 0 patients. Tabulate your results.
(c) What is the probability of having eight patients express a preference for drug A if drugs A and B were truly equal in their effects? What is the probability of having eight or more patients express a preference for drug A under the same condition? Show how these probabilities can be obtained from your table in part (b).
(d) Draw your conclusions with regard to the finding that eight patients preferred drug A. Give the assumptions you have made in carrying out the above calculations and discuss their appropriateness
Question 4
When testing subjects for HIV in surveillance studies WHO protocol states that subjects whose test result is negative on the first test should be classified as 'HIV negative'. Subjects whose test result is positive on a first test should be retested twice and classified as 'HIV positive' if one or both of these retests are positive. If both of the retests are negative then the subject is classified as 'HIV negative'.
Let the sensitivity of a single test be a and the specificity be p. Write down, in terms of a and ff. the sensitivity and specificity of the tests carried out under the WHO protocol. Use these expressions to compare the effects of using the WHO protocol as opposed to using a single test Discuss some of the advantages and disadvantages of the WHO protocol versus the single test. Give the assumptions that you have made in carrying out calculations.