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- From: Mathematics, Statistics and Probability
- Posted on: Wed 20 Jan, 2016
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1. The following are body mass index (BMI) scores measured in 12 patients who are free of diabetes and participating in a study of risk factors for obesity. Body mass index is measured as the ratio of weight in kilograms to height in meters squared. Generate a 95% confidence interval estimate of the true BMI.
25 27 31 33 26 28 38 41 24 32 35 40
2. Consider the data in Problem #1. How many subjects would be needed to ensure that a 95% confidence interval estimate of BMI had a margin of error not exceeding 2 units?
3. A clinical trial is run to investigate the effectiveness of an experimental drug in reducing preterm delivery to a drug considered standard care and to placebo. Pregnant women are enrolled and randomly assigned to receive either the experimental drug, the standard drug or placebo. Women are followed through delivery and classified as delivering preterm (< 37 weeks) or not. The data are shown below.
Preterm Delivery Experimental Drug Standard Drug Placebo
Yes 17 23 35
No 83 77 65
Is there a statistically significant difference in the proportions of women delivering preterm among the three treatment groups? Run the test at a 5% level of significance.
4. Consider the data presented in problem #4. Previous studies have shown that approximately 32% of women deliver prematurely without treatment. Is the proportion of women delivering prematurely significantly higher in the placebo group? Run the test at a 5% level of significance.
5. A study is run comparing HDL cholesterol levels between men who exercise regularly and those who do not. The data are shown below.
Regular Exercise N Mean Std Dev
Yes 35 48.5 12.5
No 120 56.9 11.9
Generate a 95% confidence interval for the difference in mean HDL levels between men who exercise regularly and those who do not.
6. A clinical trial is run to assess the effects of different forms of regular exercise on HDL levels in persons between the ages of 18 and 29. Participants in the study are randomly assigned to one of three exercise groups - Weight training, Aerobic exercise or Stretching/Yoga – and instructed to follow the program for 8 weeks. Their HDL levels are measured after 8 weeks and are summarized below.
Exercise Group N Mean Std Dev
Weight Training 20 49.7 10.2
Aerobic Exercise 20 43.1 11.1
Stretching/Yoga 20 57.0 12.5
Is there a significant difference in mean HDL levels among the exercise groups? Run the test at a 5% level of significance.
7. Consider again the data in problem #6. Suppose that in the aerobic exercise group we also measured the number of hours of aerobic exercise per week and the mean is 5.2 hours with a standard deviation of 2.1 hours. The sample correlation is -0.42.
a) Is there evidence of a significant correlation between number of hours of exercise per week and HDL cholesterol level? Run the test at a 5% level of significance.
b) Estimate the equation of the regression line that best describes the relationship between number of hours of exercise per week and HDL cholesterol level (Assume that the dependent variable is HDL level).
c) Estimate the HDL level for a person who exercises 7 hours per week.
d) Estimate the HDL level for a person who does not exercise.
8. The following data were collected in a clinical trial to compare a new drug to a placebo for its effectiveness in lowering total serum cholesterol. Generate a 95% confidence interval for the difference in mean total cholesterol levels between treatments.
New Drug
(n=75)
Placebo
(n=75)
Total Sample
(n=150)
Mean (SD) Total Serum Cholesterol 185.0 (24.5) 204.3 (21.8) 194.7 (23.2)
% Patients with Total Cholesterol < 200 78.0% 65.0% 71.5%
9. A small pilot study is conducted to investigate the effect of a nutritional supplement on total body weight. Six participants agree to take the nutritional supplement. To assess its effect on body weight, weights are measured before starting the supplementation and then after 6 weeks. The data are shown below. Is there a significant increase in body weight following supplementation? Run the test at a 5% level of significance.
Subject Initial Weight Weight after 6 Weeks
1 155 157
2 142 145
3 176 180
4 180 175
5 210 209
6 125 126
10. Answer True or False to each of the following
a) The margin of error is always greater than or equal to the standard error.
b) If a test is run and p=0.0356, then we can reject H0 at ?=0.01.
c) If a 95% CI for the difference in two independent means is (-4.5 to 2.1), then the point estimate is -2.1.
d) If a 95% CI for the difference in two independent means is (2.1 to 4.5), there is no significant difference in means
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