In a randomised comparative experiment on the effect of calcium in the diet on blood pressure, a group of researchers recruited 26 healthy males and randomly assigned them into one of two groups. One group received calcium; the other a placebo. At the beginning of the study, the researchers measured several variables on the subjects. For a research of this nature, it is important that the differences between the subjects of these groups, with respect to the variables of interest, are not overly large and unreasonable prior to the commencement of the study. One of these variables was systolic blood pressure (SBP; mm Hg). The table below provides the baseline SBP measurements for the subjects in the two groups. Placebo Calcium 112 112 103 104 117 104 127 115 126 114 131 120 95 104 113 107 125 106 105 96 108 107 99 103 98 103 (a) Find the sample mean and sample standard deviation of the SBP readings for the Placebo and Calcium groups. [2 marks] (b) Determine the 5-number summary individually for the Placebo and Calcium groups and hence, provide the corresponding side-by-side boxplots to compare the two groups. Comment as to whether you believe the subjects are comparable at baseline with respect to their SBP readings and why. [4 marks] (c) Perform the appropriate test at the 5% level of significance to determine whether there is a significant difference in the mean SBP between the Placebo and Calcium groups. State all assumptions associated with you chosen test. [5 marks] (d) Find the 95% confidence interval for the true mean difference in SBP between the Placebo and Calcium groups. [2 marks] (e) Does your 95 % confidence interval in part (c) support your conclusion in (b)? How? [2 marks] Page | 5 QUESTION 2 [10 MARKS] Makers of generic drugs must show that they do not differ significantly from the "reference" drugs that they imitate. One aspect in which they might differ is their extent of absorption in the blood. The table below gives the absorption data taken from 20 healthy non-smoking male subjects. The data was acquired from a pharmaceutical company that wishes to test a generic brand against one of their own. In the study, Subjects 1-10 received the generic drug first and Subjects 11-20 received the reference drug first. In all cases, a washout period separated the two drugs so that the first had disappeared from the blood before the subject took the second drug. Subject Reference Generic Subject Reference Generic 1 4108 1755 11 2344 2738 2 2526 1138 12 1894 2302 3 2779 1613 13 1022 1284 4 3852 2254 14 2256 3052 5 1833 1310 15 1038 1287 6 2463 2120 16 1339 1930 7 2059 1851 17 1262 1964 8 1709 1878 18 1438 2549 9 1829 1682 19 1735 3340 10 2591 2613 20 1020 3050 (a) Find the sample mean and standard deviation of the extent of absorption for the Reference and Generic drugs. [2 marks] (b) Use the appropriate test to determine whether on average, the Generic drug has a higher extent of absorption as compared to the Reference drug. State all assumptions associated with you chosen test. [4 marks] (c) Find the 95% confidence interval for the true mean difference in extent of absorption between the Generic and Reference drugs. [2 marks] (d) Does your 95 % confidence interval in part (c) support your conclusion in (b)? How? [2 marks] Page | 6 QUESTION 3 [10 MARKS] A study was conducted to examine subjects with chronic, non-specific lower back pain. In this study, 10 subjects completed a survey questionnaire assessing physical functioning at baseline, and 1, 3, and 6 months after a sham treatment on their lower backs. Their data are given in the table below. Higher values indicate better physical functioning. The goal of the experiment was to determine if the subjects would report improvement even though the treatment they received would provide minimal improvement. Subject Physical Functioning Scores Baseline 1 Month 3 Months 6 Months 1 80 60 95 100 2 95 90 95 95 3 65 55 50 45 4 50 45 70 70 5 60 75 80 85 6 70 70 75 70 7 80 80 85 80 8 70 60 75 65 9 80 80 70 65 10 65 30 45 60 (a) Find the sample mean and standard deviation of the physical functioning scores at baseline, and 1, 3 and 6 months after the sham treatment. [3 marks] (b) Use the appropriate test and determine whether there is a difference in physical functioning scores at baseline, and 1, 3 and 6 months after the sham treatment. Is there evidence of a "placebo" effect in the study? [5 marks] (c) What are the assumptions associated with your chosen test in part (b) and what tests can one use to verify these assumptions? [2 marks] Note: Testing of the assumptions is not necessary