Chapters 5, 6 and 7 …

Chapter 5

1. Total cholesterol in children aged 10-15 is assumed to follow a normal distribution with a mean of 191 and a standard deviation of 22.4.

a. What proportion of children 10-15 years of age have total cholesterol between 180 and 190?

b. What proportion of children 10-15 years of age would be classified as hyperlipidemic (Assume that hyperlipidemia is defined as a total cholesterol level over 200)?

c. What is the 90th percentile of cholesterol?

2. Among coffee drinkers, men drink a mean of 3.2 cups per day with a standard deviation of 0.8 cups. Assume the number of coffee drinks per day follows a normal distribution.

a. What proportion drink 2 cups per day or more?

b. What proportion drink no more than 4 cups per day?

c. If the top 5% of coffee drinkers are considered heavy coffee drinkers, what is the minimum number of cups consumed by a heavy coffee drinker? Hint: Find the 95th percentile.

3. A study is conducted to assess the impact of caffeine consumption, smoking, alcohol consumption and physical activity on cardiovascular disease. Suppose that 40% of participants consume caffeine and smoke. If 8 participants are evaluated, what is the probability that:

a. Exactly half of them consume caffeine and smoke?

b. At most 6 consume caffeine and smoke?

4. A recent study of cardiovascular risk factors reported that 30% of adults met criteria for hypertension. If 15 adults are assessed, what is the probability that

a. Exactly 5 meet the criteria for hypertension?

b. None meet the criteria for hypertension?

c. Less than or equal to 7 meet the criteria for hypertension? 

5. Diastolic blood pressures are assumed to follow a normal distribution with a mean of 85 and a standard deviation of 12.

a. What proportion of people have diastolic blood pressure less than 90?

b. What proportion have diastolic blood pressures between 80 and 90?

c. If someone has a diastolic blood pressure of 100, what percentile is he/she in?

Chapter 6.

1. A study is run to estimate the mean total cholesterol level in children 2-6 years of age. A sample of 9 participants is selected and their total cholesterol levels are measured as follows. 

185 225 240 196 175 180 194 147 223

Generate a 95% confidence interval for the true mean total cholesterol levels in adults with a history of hypertension.

2. A clinical trial is planned to compare an experimental medication designed to lower blood pressure to a placebo. Before starting the trial, a pilot study is conducted involving 10 participants. The objective of the study is to assess how systolic blood pressure changes over time untreated. Systolic blood pressures are measured at baseline and again 4 weeks later. Compute a 95% confidence interval for the difference in blood pressures over 4 weeks.

Baseline 120  145 130 160 152  143 126 121 115 135 

4 Weeks 122  142 135 158 155 140 130 120 124 130

3. After the pilot study (described in #2), the main trial is conducted and involves a total of 200 patients. Patients are enrolled and randomized to receive either the experimental medication or the placebo. The data shown below are data collected at the end of the study after 6 weeks on the assigned treatment.

Experimental (n=100) Placebo (n=100)

% Hypertensive  14% 22%

Generate a 95% confidence interval for the difference in proportions of patients with hypertension between groups.

4. The following data were collected as part of a study of coffee consumption among male and female undergraduate students. The following reflect cups per day consumed:

Male 3 4 6 3 2 1 0 2

Female 5 3 1 2 0 4 3 1 

Generate a 95% confidence interval for the difference in mean numbers of cups of coffee consumed between men and women. 

5. A clinical trial is conducted comparing a new pain reliever for arthritis to a placebo. Participants are randomly assigned to receive the new treatment or a placebo. The outcome is pain relief within 30 minutes. The data are shown below.

Pain Relief No Pain Relief

New Medication 44 76

Placebo  21 99

a. Generate a 95% confidence interval for the proportion of patients on the new medication who report pain relief

b. Generate a 95% confidence interval for the difference in proportions of patients who report pain relief.

Chapter 7

1. The following data were collected in a clinical trial evaluating a new compound designed to improve wound healing in trauma patients. The new compound was compared against a placebo. After treatment for 5 days with the new compound or placebo the extent of wound healing was measured and the data are shown below.

Percent Wound Healing

Treatment 0-25% 26-50% 51-75% 76-100%

New Compound (n=125) 15 37 32 41

Placebo (n=125) 36 45 34 10

Is there a difference in the extent of wound healing by treatment? (Hint: Are treatment and the percent wound healing independent?) Run the appropriate test at a 5% level of significance. 

2. Use the data in Problem #1 and pool the data across the treatments into one sample of size n=250. Use the pooled data to test whether the distribution of the percent wound healing is approximately normal. Specifically, use the following distribution: 30%, 40%, 20% and 10% and α =0.05 to run the appropriate test.

3. The following data were collected in an experiment designed to investigate the impact of different positions of the mother on fetal heart rate. Fetal heart rate is measured by ultrasound in beats per minute. The study included 20 women who were assigned to one position and had the fetal heart rate measured in that position. Each woman was between 28-32 weeks gestation. The data are shown below. 

Back Side Sitting Standing

20 21 24 26

24 23 25 25

26 25 27 28

21 24 28 29

19 16 24 25

Is there a significant difference in mean fetal heart rates by position? Run the test at a 5% level of significance. 

4. A clinical trial is conducted comparing a new pain reliever for arthritis to a placebo. Participants are randomly assigned to receive the new treatment or a placebo and the outcome is pain relief within 30 minutes. The data are shown below.

Pain Relief No Pain Relief
New Medication 44 76
Placebo  21 99

Is there a significant difference in the proportions of patients reporting pain relief? Run the test at a 5% level of significance.

5. A clinical trial is planned to compare an experimental medication designed to lower blood pressure to a placebo. Before starting the trial, a pilot study is conducted involving 7 participants. The objective of the study is to assess how systolic blood pressure changes over time untreated. Systolic blood pressures are measured at baseline and again 4 weeks later Is there a statistically significant difference in blood pressures over time? Run the test at a 5% level of significance.

Baseline 120 145 130 160 152  143 126 

4 Weeks 122 142 135 158 155 140 130 

6. A hypertension trial is mounted and 12 participants are randomly assigned to receive either a new treatment or a placebo. Each participant takes the assigned medication and their systolic blood pressure (SBP) is recorded after 6 months on the assigned treatment. The data are as follows. 

Placebo New Treatment

134 114

143 117

148 121

142 124

150 122

160 128

Is there a difference in mean SBP between treatments? Run the appropriate test at α =0.05.