Posted: December 15th, 2016

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?

Question

Test Bank – Final Exam

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. The mean BMI in patients free of diabetes was reported as 28.2. The investigator conducting the study described in Problem 1 hypothesizes that the BMI in patients free of diabetes is higher. Based on the data in Problem 1 is there evidence that the BMI is significantly higher that 28.2? Use a 5% level of significance.

4. Peak expiratory flow (PEF) is a measure of a patient’s ability to expel air from the lungs. Patients with asthma or other respiratory conditions often have restricted PEF. The mean PEF for children free of asthma is 306. An investigator wants to test whether children with chronic bronchitis have restricted PEF. A sample of 40 children with chronic bronchitis are studied and their mean PEF is 279 with a standard deviation of 71. Is there statistical evidence of a lower mean PEF in children with chronic bronchitis? Run the appropriate test ata=0.05.

5. Consider again the study in Problem 4, a different investigator conducts a second study to investigate whether there is a difference in mean PEF in children with chronic bronchitis as compared to those without. Data on PEF are collected and summarized below. Based on the data, is there statistical evidence of a lower mean PEF in children with chronic bronchitis as compared to those without? Run the appropriate test ata=0.05.

Group

Number of Children

Mean PEF

Std Dev PEF

Chronic Bronchitis

25

281

68

No Chronic Bronchitis

25

319

74

6. Using the data presented in Problem 5,

a) Construct a 95% confidence interval for the mean PEF in children without chronic bronchitis.

b) How many children would be required to ensure that the margin of error in (a) does not exceed 10 units?

7. 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. 8. Using the data in Problem 7, generate a 95% confidence interval for the difference in proportions of women delivering preterm in the experimental and standard drug treatment groups. 9. Consider the data presented in Problem 7. 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. 10. 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.

11. 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. HINT: SSerror = 7286.5.

12. Consider again the data in Problem 11. 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) 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).

b) Estimate the HDL level for a person who exercises 7 hours per week.

c) Estimate the HDL level for a person who does not exercise.

13. The table below summarizes baseline characteristics on patients participating in a clinical trial.

Characteristic

Placebo (n=125)

Experimental (n=125)

P

Mean (+SD) Age

54 + 4.5

53 + 4.9

0.7856

% Female

39%

52%

0.0289

% Less than High School Education

24%

22%

0.0986

% Completing High School

37%

36%

% Completing Some College

39%

42%

Mean (+SD) Systolic Blood Pressure

136 + 13.8

134 + 12.4

0.4736

Mean (+SD) Total Cholesterol

214 + 24.9

210 + 23.1

0.8954

% Current Smokers

17%

15%

0.5741

% with Diabetes

8%

3%

0.0438

a) Are there any statistically significant differences in baseline characteristics between treatment groups? Justify your answer.

b) Write the hypotheses and the test statistic used to compare ages between groups. (No calculations – just H0, H1 and form of the test statistic)

c) Write the hypotheses and the test statistic used to compare % females between groups. (No calculations – just H0, H1 and form of the test statistic)

d) Write the hypotheses and the test statistic used to compare educational levels between groups. (No calculations – just H0, H1 and form of the test statistic)

14. A study is designed to investigate whether there is a difference in response to various treatments in patients with rheumatoid arthritis. The outcome is patient’s self-reported effect of treatment. The data are shown below. Is there a significant difference in effect of treatment? Run the test at a 5% level of significance.

Symptoms

Worsened

No Effect

Symptoms Improved

Total

Treatment 1

22

14

14

50

Treatment 2

14

15

21

50

Treatment 3

9

12

29

50

15. Using the data shown in Problem 14, suppose we focus on the proportions of patients who show improvement. Is there a statistically significant difference in the proportions of patients who show improvement between treatments 1 and 2. Run the test at a 5% level of significance.

16. An analysis is conducted to compare mean time to pain relief (measured in minutes) under four competing treatment regimens Summary statistics on the four treatments are shown below.

Treatment

Sample Size

Mean Time to Relief

Sample Variance

A

5

33.8

17.7

B

5

27.0

15.5

C

5

50.8

9.7

D

5

39.6

16.8

a) Complete the following ANOVA Table

Source of Variation

SS

df

MS

F

Between Groups

Within Groups

3719.48

Total

b) Write the hypotheses to be tested.

c) Write the decision rule.

d) What is the conclusion?

17. 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% 18. Using the data in Problem 17, a) Generate a 95% confidence interval for the proportion of all patients with total cholesterol < 200. b) How many patients would be required to ensure that a 95% confidence interval has a margin of error not exceeding 5%? 19. 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 20. The following table was presented in an article summarizing a study to compare a new drug to a standard drug and to a placebo. Characteristic* New Drug Standard Drug Placebo p Age, years 45.2 (4.8) 44.9 (5.1) 42.8 (4.3) 0.5746 % Female 51% 55% 57% 0.1635 Annual Income, $000s 59.5 (14.3) 63.8 (16.9) 58.2 (13.6) 0.4635 % with Insurance 87% 65% 82% 0.0352 Disease Stage 0.0261 Stage I 35% 18% 33% Stage II 42% 37% 47% Stage III 23% 51% 20% *Table entries and Mean (SD) or % a) Are there any statistically significant differences in the characteristics shown among the treatments? Justify your answer. b) Consider the test for differences in age among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only). c) Consider the test for differences in insurance coverage among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only). d) Consider the test for differences in disease stage among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only). 21. A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are randomly assigned to receive either the new drug or the currently available drug and report improvement in pain on a 5-point ordinal scale: 1=Pain is much worse, 2=Pain is slightly worse, 3= No change, 4=Pain improved slightly, 5=Pain much improved. Is there a significant difference in self-reported improvement in pain? Use the Mann-Whitney U test with a 5% level of significance. New Drug: 4 5 3 3 4 2 Standard Drug: 2 3 4 1 2 3 22. 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 ata=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. e) If a 90% CI for the mean is (75.3 to 80.9), we would reject H0:m=70 in favor of H1: m?70 ata=0.05. 23. A randomized controlled trial is run to evaluate the effectiveness of a new drug for asthma in children. A total of 250 children are randomized to either the new drug or placebo (125 per group). The mean age of children assigned to the new drug is 12.4 with a standard deviation of 3.6 years. The mean age of children assigned to the placebo is 13.0 with a standard deviation of 4.0 years. Is there a statistically significant difference in ages of children assigned to the treatments? Run the appropriate test at a 5% level of significance. 30. For each statement below, indicate whether the statement is true or false. a) In logistic regression, the predictors are dichotomous, and the outcome is a continuous variable. b) When calculating a correlation coefficient between two continuous variables, the scales on which the variables are measured affect the value of the correlation coefficient. c) It is more difficult to reject a null hypothesis if we use a 10% level of significance compared with a 5% level of significance. d) The sample size required to detect an effect size of 0.25 is larger than the sample size required to detect an effect size of 0.50 with 80% power and a 5% level of significance. 31. For each question below, provide a brief (1-2 sentences) response. a) How is the slope coefficient (b1) in a simple linear regression different than the coefficient (b1) in a multiple linear regression model? b) When would a survival analysis model be used instead of a logistic regression model? c) What is the appropriate statistical test to assess whether there is an association between obesity status (normal weight, overweight, obese) and 5-year incident cardiovascular disease (CVD)? Suppose each participant’s obesity status (category) is known as is whether they develop CVD over the next 5 years or not.