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 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.
- 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 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:
- 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).
