1 BMI is measured in a sample of patients, with a mean of 31.7 and a standard deviation of 5.9. 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?

2 The mean BMI in patients free of diabetes was elsewhere reported as 28.2. The researcher producing the data in question 1 wonders if the BMI in his patients free of diabetes is higher than this reported number. 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.

3 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 at a=0.05.

4 Considering again the study in Problem 3, 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 at a=0.05.

5 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. 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.

6 Consider the data presented in Problem 5. 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.

7 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. Is there a significant difference in mean HDL levels among the exercise groups? Run the test at a 5% level of significance.

8 Consider again the data in Problem 7. 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 equation describing the results is Y = 54.6 - 2.22x, where Y = HDL levels.

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

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

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

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)

10 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.

11 Using the data shown in Problem 10, 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.

12 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. Test the null hypothesis that there is no difference in mean time to pain relief between treatment groups.

13 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.

14 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.

15 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 a=0.01.

16 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.

17 "Average adult Americans are about one inch taller, but nearly a whopping 25 pounds heavier than they were in 1960, according to a new report from the Centers for Disease Control and Prevention (CDC).

The bad news, says CDC is that average BMI (body mass index, a weight-for-height formula used to measure obesity) has increased among adults from approximately 25 in 1960 to 28 in 2002."

Boston is considered one of America's healthiest cities - is the weight gain since 1960 similar in Boston? A sample of n=25 adults suggested a mean increase of 17 pounds with a standard deviation of 8.6 pounds. Is Boston statistically significantly different in terms of weight gain since 1960?

Run the appropriate test at a 5% level of significance.

18 In 2007, the CDC reported that approximately 6.6 per 1000 (0.66%) children were affected with autism spectrum disorder. A sample of 900 children from Boston are tested and 7 are diagnosed with autism spectrum disorder. Is the proportion of children affected with autism spectrum disorder higher in Boston as compared to the national estimate? Run the appropriate test at a 5% level of significance.

19 For each question below, provide a brief (1-2 sentences) response.

a When would a survival analysis model be used instead of a logistic regression model?

b 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.

20 An observational study is conducted to compare experiences of men and women between the ages of 50-59 years following coronary artery bypass surgery.

Participants undergo the surgery and are followed until the time of death, until they are lost to follow-up or up to 30 years, whichever comes first.

The following table details the experiences of participating men and women. The data below are years of death or years of last contact for men and women.

Estimate the survival functions for each treatment group (men, women) using the Kaplan-Meier approach