Analysis of Scientific Data - Semester 2, 2023 Workshop 5 - Exercises Part A - Normal Distribution Suppose that pulse rates while completing the student survey come from a Normal distribution with mean 71.7 bpm and standard deviation 11.7 bpm. a) What is the probability that a random student has a pulse rate of at least 90 bpm? b) What is the probability that a random student has pulse rate between 60 and 80 bpm? c) What value would put a student in the bottom 10% of pulse rates? ! d) In a random sample of 5 students, what is the probability that at least 3 of them have a pulse rate over 90 bpm while completing the survey?
Part B - 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 Drug B Difference 11 ! 74 55 63 58 -3 61 49 12 47 53 41 50 6 3 74 52 40 69 59 5 -7 50 31 44 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 p, 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? You may find the following output from R useful in answering the questions on this sheet. ! ! > sum(dbinom(x=3:5,size=5,prob =. 0589)) > pt(0.7454,df=8) [1] 0.001867087 > sum(dbinom(x=3:5,size=5,prob =. 1142)) [1] 0.01245883 > pnorm(-2.114) [1] 0.01725763 > qt(.95,df=8) > pnorm(1.564) [1] 1.859548 [1] 0.9410912 > qt(.975,df=8) [1] 2.306004 > pnorm(-1.564) > pt(2.236,df=8) [1] 0.05890878 [1] 0.9721138 > pnorm(0.7094017) [1] 0.7609624 > pnorm(-1) [1] 0.1586553 > qnorm(.9) [1] 1.281552 > qnorm(.1) [1] 0.7613218 > 1-pt(2.236,df=8) [1] 0.02788622 > pt(2.236,df=9) [1] 0.9739085 > 0.0589^3 [1] 0.0002043365 [1] -1.281552