Chronic obstructive pulmonary disease (COPD) is a type of obstructive lung disease characterized by long-term breathing problems and poor airflow.
statistics
Description
For the normally-distributed random variable X, find:
The numerical value of k such that P(μ – kσ ≤ X ≤ μ + kσ) = 0.8502.
The numerical value of k where P (X ≤ k) = 0.2358, given μ =100 and σ = 15.
The numerical value of μ, given σ = 10 and P(X ≤ 40) =0.2643.
The numerical value of σ, given μ = 30 and P(X ≤ 50) = 0.9686.
Chronic obstructive pulmonary disease (COPD) is a type of obstructive lung disease characterized by long-term breathing problems and poor airflow. COPD is a progressive disease, meaning lung obstruction worsens over time. The degree of obstruction is measured by forced expiratory volume (FEV), an index of pulmonary function that measures the volume of air expelled after 1 second of constant effort. FEV is typically normally distributed and affected by age and sex. Assume that 40 year old men without COPD have an FEV with mean = 5.0 L and standard deviation = 0.6 L. In comparably aged men with COPD, assume that FEV has a mean = 3.5 L and standard deviation = 0.9 L. Use these data to answer the questions below.
An FEV of less than 3.0 L is regarded as showing moderate functional impairment (e.g., periodic breathlessness, difficulty climbing stairs). What is the probability that a 40-year old man with COPD has moderate functional impairment?
Answer question a for a 40-year old man without COPD.
In a population of 25,000 men age 40 years and without COPD, how many would you expect to have functional impairment?
In a population of 2,500 men age 40 years with COPD, how many would you expect to be free of functional impairment?
Some people with COPD are not functionally impaired now, but because pulmonary function declines with age in people with COPD they may become functionally impaired at some future point. Assume that the average annual decline in FEV is 0.03 L for men with COPD and that there is no change in the variance over time. What is the probability that a 40-year old man with COPD will be functionally impaired by age 70?
Scores made on an aptitude test by medical students are approximately normally distributed with a mean of 500 and a variance of 10,000. Use this information to answer the following questions.
What proportion of those taking the test score below 200?
A person is about to take the test. What is the probability that he or she will obtain a score of 650 or more?
What proportion of scores fall between 350 and 650?
The probability of being below a certain score was calculated to be 11.12%. What was the score corresponding to this probability?
The Diabetes Prevention Trial (DPT) was a large, country-wide, weight-loss trial in which half the participants received an active 24-month intervention consisting of bi-weekly group sessions and individual counselling meetings. The other half of the DPT participants received a control intervention. For participants in the active intervention group, the average reduction in body mass index (BMI, i.e., weight in kg/height2 in m2) over 24 months was 1.69 kg/m2. The standard deviation of change in BMI was 6.7 kg/m2. In the control group of the DPT, the mean change in BMI was 0 units with a standard deviation of 6 kg/m2. Assume that the distribution of BMI change in both groups is approximately normal. Use these data to answer the questions below.
What is the probability that a random active group participant would lose at least 1 BMI unit over 24 months?
What is the probability that a random control group participant would lose at least 1 BMI unit over 24 months?
In a population of 850,000 people that received the active intervention, how many people would you expect to lose at least 2 BMI units over 24 months?
It was later discovered that only 70% of DPT participants in the active group actually complied with the intervention, that is, 30% of the active group participants either dropped out or did not attend the required group and individual counselling sessions. This drop out group had the same distribution of change as the participants in the control group. In light of this new information, in a population of 850,000 people enrolled in the active intervention, how many would you expect to lose at least 2 BMI units over 24 months?
References:
The questions in this tutorial assignment were adapted from practice questions from Chapter 5 of the textbook by Rosner, B. (2011). Fundamentals of Biostatistics, 7th Edition. Brooks/Cole Cengage Learning, Boston, MA.
