Problem 1:
(i) In a study of uptake of maternal vaccinations by Indigenous women in Central Australia (Krishnaswamy et al., Aust N Z J Public Health, 2018), 69.6% of the participating women (i.e. 32 out of 46) received an influenza vaccine while they were pregnant.
(a) If six women are randomly selected from those who took part in this study, what is the probability that exactly one of the six will have been vaccinated against influenza while pregnant?
(b) What is the probability that at least one of the six women will have received an influenza vaccine while pregnant?
(c) In answering the above Problems, what assumption(s) were you making?
(d) How many (of the six) women would be expected to have received an influenza vaccine while pregnant?
(ii) In a study to investigate the effectiveness of ultrasound screening for detecting developmental dysplasia of the hip (DDH) in newborns by Roovers et al. (Arch Dis Child Fetal Neonatal Ed, 2005) sensitivity of 88.5% was reported. The specificity of ultrasound screening for DDH is thought to be 96.7%.
If the prevalence of DDH in the target (newborn) population is assumed to be 5.2%:
(a) Using the information presented above, calculate the probability that a newborn in the target population without DDH is classified incorrectly by ultrasound screening as having DDH.
(b) Calculate the probability that a newborn in the target population with a positive classification of DDH by ultrasound screening is classified correctly.
If the prevalence of DDH in the newborn population is actually equal to 0.52%:
(c) Calculate the probability that a newborn in the target population with a positive classification of DDH (by ultrasound screening) is classified correctly.
(d) Based on your answers to (a) through (c) above, what conclusions do you draw about this screening test?
Problem 2:
This Problem is designed to give you some practice in looking up Statistical Tables. Remember to draw diagrams to make sure you are on the right track. Be careful to consider the effect of any absolute value signs, i.e. | |. State the following probability values - either exact values or the range of possible values indicated on the Statistical Tables. For part (g), work from the stated probability to determine the value of a.
(a) P(z < 1.96)
(b) P(z > -1.96)
(c) P(x > 2) (assume that mean of the Normal distribution of the random
variable x is 2/3 and the variance is 16) (d) P(| z | < 1.645)
(e) If P(z > a) = 0.40, what value would a need to take?
Problem 3:
In a study on the effectiveness of Ginkgo biloba in treating tinnitus, 24 participants were recruited through advertisements in the national press in the United Kingdom. Once enrolled in the study, all participants were asked to complete a number of Problemnaires that allowed for the calculation of a severity of tinnitus score. Participants were then instructed to take three tablets per day (each containing 50mg of Ginkgo biloba) over a 12 week period. After this time all participants completed the same Problemnaires as on enrolment, so that their severity of tinnitus could again be calculated. All participants were assumed to have complied with the treatment regimen.
The mean (and standard deviation) of the severity of tinnitus scores was calculated for patients at entry to the study and again after the 12 week period.
(a) Define the sample statistics of interest in this study.
(b) Define the parameters of interest in this study.
(c) State an appropriate null and alternative hypothesis for this study.
Problem 4:
A paper by Holland et al. (2005) [Does home based medication review keep older people out of hospital?: The HOMER randomised controlled trial. British Medical Journal 330(7486):293] concerned the reporting of evidence obtained by a randomised controlled trial that investigated whether an in-home medication review reduced the number of unplanned re-admissions to hospital.
The participants were patients aged 80 years or more, who had experienced an emergency admission to hospital (for any cause), were prescribed two or more drugs on discharge, and were returning to their own home or residential care on discharge.
Participants were randomised to receive either the intervention, which comprised two home visits by a pharmacist within two weeks and eight weeks of discharge to educate and aid patients with their medications, or the control, which comprised usual care.
The statistical analysis focused on 415 individuals randomised to the intervention and 414 individuals randomised to control. The primary outcome measure was the number of emergency re-admissions to hospital at 6 months.
The average number of re-admissions (and standard deviation) was calculated for participants randomised to the intervention and for participants randomised to control.
(a) Define the sample statistics of interest in this study.
(b) Define the parameters of interest in this study.
(c) State an appropriate null and alternative hypothesis for this study.