**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.

- What proportion of children 10-15 years of age have total cholesterol between 180 and 190?
- 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)?
- What is the 90
^{th}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.

- What proportion drink 2 cups per day or more?
- What proportion drink no more than 4 cups per day?
- 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 95
^{th}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:

- Exactly half of them consume caffeine and smoke?
- 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

- Exactly 5 meet the criteria for hypertension?
- None meet the criteria for hypertension?
- 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.

- What proportion of people have diastolic blood pressure less than 90?
- What proportion have diastolic blood pressures between 80 and 90?
- If someone has a diastolic blood pressure of 100, what percentile is he/she in?

**Chapter 6.**

Probl. 1. (In unit 6, practice problem 1 there is a mistake in the question. Instead of “ adults” work the problem for “children”.

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

Generate a 95% confidence interval for the true mean total cholesterol level in children. 6.6 Practice Problems

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 ?=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 ?=0.05.

**Chapter 8**

1. Suppose we want to design a new placebo-controlled trial to evaluate an experimental medication to increase lung capacity. The primary outcome is peak expiratory flow rate, a continuous variable measured in liters per minute. The primary outcome will be measured after 6 months on treatment. The expected peak expiratory flow rate in adults is 300 with a standard deviation of 50. How many subjects should be enrolled to ensure 80% power to detect a difference of 15 liters per minute with a two sided test and ?=0.05?

2. An investigator wants to estimate caffeine consumption in high school students. How many students would be required to ensure that a 95% confidence interval estimate for the mean caffeine intake (measured in mg) is within 15 units of the true mean? Assume that the standard deviation in caffeine intake is 68 mg.

3. Consider the study proposed in problem #2. How many students would be required to estimate the proportion of students who consume coffee? Suppose we want the estimate to be within 5% of the true proportion with 95% confidence.

4. A clinical trial was conducted comparing a new compound designed to improve wound healing in trauma patients to a placebo. After treatment for 5 days, 58% of the patients taking the new compound had a substantial reduction in the size of their wound as compared to 44% in the placebo group. The trial failed to show significance. How many subjects would be required to detect the difference in proportions observed in the trial with 80% power? A two sided test is planned at ?=0.05.

5. A crossover trial is planned to evaluate the impact of an educational intervention program to reduce alcohol consumption in patients determined to be at risk for alcohol problems. The plan is to measure alcohol consumption (the number of drinks on a typical drinking day) before the intervention and then again after participants complete the educational intervention program. How many participants would be required to ensure that a 95% confidence interval for the mean difference in the number of drinks is within 2 drinks of the true mean? Assume that the standard deviation of the difference in the mean number of drinks is 6.7 drinks.

6. An investigator wants to design a study to estimate the difference in the proportions of men and women who develop early onset cardiovascular disease (defined as cardiovascular disease before age 50). A study conducted 10 years ago, found that 15% and 8% of men and women, respectively, developed early onset cardiovascular disease. How many men and women are needed to generate a 95% confidence interval estimate for the difference in proportions with a margin of error not exceeding 4%?

7. The mean body mass index (BMI) for boys age 12 is 23.6. An investigator wants to test if the BMI is higher in boys age 12 living in New York City. How many boys are needed to ensure that a two-sided test of hypothesis has 80% power to detect an increase in BMI of 2 units? Assume that the standard deviation in BMI is 5.7.

8. An investigator wants to design a study to estimate the difference in the mean BMI between boys and girls age 12 living in New York City. How many boys and girls are needed to ensure that a 95% confidence interval estimate for the difference in mean BMI between boys and girls has a margin of error not exceeding 2 units? Use the estimate of the variability in BMI from problem #7.

**Chapter 9**

1. Consider the following data measured in a sample of n=25 undergraduates in an on-campus survey of health behaviors. Enter the data into an Excel worksheet for analysis.

ID | Age | Female Sex | Year in School | GPA | Current Smoker | # Hours Exercise per Week | # Average Number of Drinks per Week | # Cups Coffee per Week |

1 | 18 | 1 | Fr | 3.85 | 1 | 7 | 3 | 3 |

2 | 21 | 0 | Jr | 3.27 | 1 | 3 | 2 | 4 |

3 | 19 | 1 | So | 2.90 | 0 | 0 | 4 | 7 |

4 | 22 | 0 | Sr | 3.65 | 1 | 0 | 2 | 4 |

5 | 21 | 1 | Sr | 3.41 | 1 | 0 | 1 | 3 |

6 | 20 | 0 | Jr | 3.20 | 0 | 2 | 5 | 8 |

7 | 19 | 1 | Jr | 2.89 | 1 | 1 | 4 | 10 |

8 | 17 | 0 | Fr | 3.75 | 0 | 6 | 0 | 0 |

9 | 18 | 0 | So | 4.00 | 0 | 6 | 2 | 6 |

10 | 17 | 1 | So | 3.18 | 0 | 3 | 5 | 7 |

11 | 21 | 0 | Jr | 2.58 | 1 | 3 | 12 | 12 |

12 | 22 | 1 | Sr | 2.98 | 0 | 2 | 3 | 4 |

13 | 19 | 0 | Fr | 3.16 | 1 | 2 | 0 | 6 |

14 | 21 | 1 | Jr | 3.36 | 1 | 3 | 1 | 2 |

15 | 22 | 1 | So | 3.72 | 0 | 6 | 3 | 0 |

16 | 19 | 0 | So | 3.30 | 1 | 4 | 0 | 6 |

17 | 16 | 0 | Fr | 3.28 | 0 | 4 | 0 | 5 |

18 | 22 | 0 | Sr | 2.98 | 0 | 0 | 8 | 5 |

19 | 17 | 1 | Fr | 3.90 | 0 | 7 | 0 | 2 |

20 | 20 | 1 | Sr | 3.78 | 1 | 4 | 6 | 2 |

21 | 21 | 1 | So | 3.26 | 1 | 2 | 3 | 4 |

22 | 23 | 0 | Jr | 3.01 | 0 | 1 | 9 | 7 |

23 | 23 | 0 | Sr | 3.83 | 1 | 5 | 4 | 4 |

24 | 17 | 1 | Fr | 3.76 | 0 | 5 | 2 | 1 |

25 | 22 | 1 | Sr | 3.05 | 0 | 1 | 5 | 5 |

2. Estimate the simple linear regression equation relating number of cups of coffee per week to GPA (Consider GPA the dependent or outcome variable).

3. Estimate the simple linear regression equation relating female sex to GPA (Consider GPA the dependent or outcome variable).

4. Estimate the multiple linear regression equation relating number of cups of coffee per week, female sex and number of hours of exercise per week to GPA (Consider GPA the dependent or outcome variable).