In a study of a new drug that is designed to reduce blood pressure in a particular
In a study of a new drug that is designed to reduce blood pressure in a particular clinical population, suppose we let Y1, , Yn denote the reduction in systolic blood pressure (SBP) that is observed over a 3-month period in each member of a sample of in patients. In a preliminary study of 15 patients, the mean reduction in systolic blood pressure was found to be i = 8.1 mmHg with a standard deviation of Sn1 = 7.2 mmHg. Suppose we assume a parametric model for these data such that the reduction in SBP for each patient is considered to be normally distributed, with mean u and variance (:2, i.e. has the distribution N01, 02). Consider the following two proposed 95% condence intervals for the mean reduction in SBP that is seen in the population over 3 months after starting to take the drug: 3n1 ? i 30.975 3/- . 3n1 J7 i t 1.o.915 1/3 where n is the sample size and the percentiles of the normal and t _1 distributions are given, for the values n = 15, in this table: n=10 n=15 11:20 n=40 n=100 n=400 2035.5 ? 1.960 ? '2 ? '2 aims '2 2.145 '2 a ? a (b) [1 mark] Fill in the missing values in this table, i.e. nd the stated percentiles of each distribution (to the same number of decimal places). Make sure you show how you obtained your answers. [This can be done using the appropriate probability function in Stata or R.] [1 mark] Which of the two condence intervals will have the higher coverage probability for low sample sizes and why? (Explain without doing any calculations for this.) [2 marks] Suppose that the population mean reduction in SBP caused by this drug is actually 9.6 mmHg and that the population standard deviation of this reduction is 10.4 mmol/L (God told us!). Based on these assumptions about the population, for each of the sample sizes in the table above carry out a simulation of 10,000 studies and estimate the coverage probability of each of the condence intervals. You can do this by using the 'cnorm' Stata program (or equivalent R mction) that was provided in the code package on the LMS. (d) (f) (g) [1 mark] As a side-question, how many decimal places is it reasonable to use for the coverage probabilities that you estimated in the last question? Explain why, using an appropriate standard error calculation. [2 marks] Based on your results in part (c), explain when the condence intervals differ in their coverage probabilities and why. What are the practical implications around which condence interval is preferable? [2 marks] Going back to the particular study of interest (ignoring the hypothetical information about the true parameter values given in part (c)), what might be inferred om the data provided about the true average reduction in SBP, using an appropriate estimate and condence interval? Explain... [1 mark] The estimate 37 is the observed value of the estimator 17. What is the expectation and variance of this estimator under the specied model? Use these formulas to explain why the sample mean is an unbiased and consistent estimator of the population mean
