Analysis of Scientific Data - Semester 2, 2020 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: u=0 H1: b) Calculate the t statistic to test this null hypothesis. t9=(4.67-0)/(6.265/sqrt(9)) = 2.236233 c) What is the corresponding p-value? What do you conclude? P(T9>=2.236233) = 1-pt(2.236233,df=8) = 0.02787609 We can conclude that there is moderate evidence that drug A gave higher reductions in cholesterol than drug B. d) Based on this data, calculate a 95% confidence interval for the mean difference in cholesterol reduction between drug A and drug B. critical value t(9-1)=qt(0.975, df=9-1)=2.306004 [ x-2.306004*(6.265/sqrt9) , x+2.306004*(6.265/sqrt9) ] [ 4.67-4.815697 , 4.67+4.815697 ] [ -0.146, 9.486 ]
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? n = (1.96*6.265/2)^2 = 37.7 = 38 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: p= 0.5 H1: „ > 0.5 b) Assuming Ho is true, what is the distribution of X, the number of positive differences in 9 trials? X ~ Binomial(9,0.5) c) Based on the observed number of positive differences, what is the p-value for this test? What do you conclude? sum(dbinom(x=7:9,size=9,prob =. 5)) = 0.08984375 Not significant, weak information that drug A gives lower reduction d) How do your results compare with the results in Part A? Weaker results compared to Part A. This is because we don't use the whole data. We only look at one aspect (whether there is a positive difference), and it wastes a