Analysis of Scientific Data - Semester 1, 2021 Week 6 Tutorial Part A - Paired T Test Alice recruited nine pairs of identical twins for a study of two cholesterol-reducing drugs, A and B, with the aim of showing that drug A gave higher reductions in cholesterol than drug B. One of the twins in each pair was given drug A and the other was given drug B, where the choice was made at random. The amount by which cholesterol was reduced in each subject (mg/dL) is given in the following table: Pair 1 2 3 4 5 6 7 8 9 Drug A 74 55 61 47 53 74 52 40 50 Drug B 63 58 49 41 50 69 59 31 44 Difference 11 -3 12 6 3 5 -7 9 6 The sample mean difference in cholesterol reduction between drug A and drug B was 4.67 mg/dL with a sample standard deviation of 6.265 mg/dL. a) State the null and alternative hypotheses of interest in terms of u, the mean difference in cholesterol reduction between drug A and drug B in twins in this population. H0: H1: b) Calculate the t statistic to test this null hypothesis. c) What is the corresponding p-value? What do you conclude? d) Based on this data, calculate a 95% confidence interval for the mean difference in cholesterol reduction between drug A and drug B.
e) Suppose we wanted to carry out a new study that could estimate the mean difference in cholesterol reduction with a margin of error of 2 mg/dl. What sample size should the new study use? Part B - Sign Test A simple alternative to the t test is to count the number of twins where there was a positive difference between drug A and drug B. a) State the null and alternative hypotheses of interest in terms of p, the probability that, in a random pair of identical twins, the one with drug A will have a greater reduction in cholesterol than the one with drug B. H0: H1: b) Assuming Ho is true, what is the distribution of X, the number of positive differences in 9 trials? c) Based on the observed number of positive differences, what is the p-value for this test? What do you conclude? d) How do your results compare with the results in Part A? You may find the following output from R useful in answering the questions on this sheet. > pt(0.7454,df=8) > qt(.95,df=8) [1] 0.7613218 > 1-pt(2.2362,df=8) [1] 0.02787752 [1] 2.306004 > pt(2.2362, df=9) [1] 0.973917 > pt(2.2362, df=8) [1] 0.9721225 [1] 1.859548 > qt(.975,df=8) > sum(dbinom(x=7:9,size=9,prob =. 5)) [1] 0.08984375 > dbinom(x=7,size=9,prob =. 5) [1] 0.0703125