Reference no: EM132399803
BIOSTATS 540 - Introductory Biostatistics
Probabilities in Epidemiology, Sampling, Binomial, Normal
1. Take a few minutes to choose and reflect a bit upon a research area of your choice. Now, in 5 sentences at most, illustrate the concepts of population, sample, parameter, and statistic by an example from a research area of your choice. Suggestion: To help you in developing your answer, come up with a research question that you might be interested in investigating.
2. Pulmonary embolism is a relatively common condition that necessitates hospitalization and also often occurs in patients hospitalized for other reasons. An oxygen tension (arterial PO2) < 90 mm Hg is one of the important criteria used in diagnosing this condition. Suppose that the sensitivity of this test is 95%, the specificity is 75% and the estimated prevalence of pulmonary embolism is 20%.
2a. What is the predictive value positive of the oxygen tension test?
2b. In 1-2 sentences, explain the meaning of the predictive value positive that you obtained in #2a.
3. Consider a quality assessment of the manufacture of fuses. A consumer is interested in purchasing a box of 100 fuses. Before buying, however, he or she tests four. If the source box contains 10 defective fuses, what is the probability that they will find none in their sample of 4 fuses sample? Use an appropriately defined binomial distribution to answer this question.
4.
4a. What is the probability that a random variable assumed to be distributed normal with mean μ and variance σ2 has value between μ - 0.54σ and μ + 1.72σ?
4b. What percent of the possible outcomes of a normal random variable lie within plus or minus 0.64 standard deviations of the mean?
5. Some of the ideas of the normal probability distribution are relevant to the characterization of laboratory tests. Suppose that albumin has a normal distribution in a healthy population with mean μ = 3.75 mg/dl and σ = 0.50 mg/dl. The “normal” range of values will be defined as μ + 1.96 σ, so that values outside this interval will be classified as “abnormal”. Patients with advanced chronic liver disease have reduced albumin levels that are also normally distributed. Suppose that, in this population, the mean is μ = 2.5 mg/dl and σ = 0.50 mg/dl.
5a. In evaluating results for a single patient, what values of albumin will lead to a classification of “abnormal”?
5b. What proportion of patients in the population with advanced chronic liver disease will have values of albumin that will lean to an incorrect classification of “normal”?
5c. If a single patient with advanced chronic liver disease is tested, what is the probability that they will be correctly classified?
6. Interestingly, it is generally agreed that we are poor probabilists. Here is the question from an undisclosed source (please do not Google):
“A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys.
However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower. For a period of one year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you thing recorded more such days? The larger hospital, the smaller hospital, or where they about the same (that is within 5% of each other)? In developing your answer to this question, you are being asked to work with two binomial probability distributions, once for the larger hospital and once for the smaller hospital.
Hint: What is the expected value of each of the two binomial distributions and how do they compare?
7. Suppose we are told that a population distribution of a random variable X is normal with parameters μ = 1 and σ 2 = 9. Simple random samples of sample size n=9 are taken and the sample means Xn=9 are obtained.
7a. What is the correct probability model for the associated sampling distribution of Xn=9 ? What are the values of its mean (μ) and variance parameters (σ2)?
7b. Find Probability [ 1 < X‾n=9 2.85 ]
7c. Consider next a new random variable W = 4 X‾n=9. What is the correct normal probability model for the sampling distribution of W? Specifically, what are the values of its mean (μ) and variance parameters (σ2)? Tip! If you’re not at all sure how to do this, take a look at the supplementary notes to Unit 8.
8. Consider a randomized trial of treatment “Active” versus treatment “Standard” for a given disease. At one clinic, eight out of nine patients were randomized to receive treatment “Active”. A complaint is made that randomization must not be working, that π = Pr[treatment assignment is “ACTIVE”] cannot be equal to 0.50.
8a. Under the assumption that randomization is working, calculate the probability of obtaining 8 or more assignments to treatment “Active” out of 9 assignments total.
8b. Next consider that this is a multi-site randomized trial with 15 sites. Under the assumptions that (1) randomization is working and (2) there are 9 assignments at each of all 15 sites, calculate the probability that there will be more than one site with 8 or more assignments to “Active”.
9. Recall the definition of the interquartile range, IQR. IQR = P75 – P25.
9a. If Z is distributed standardized normal ( μ = 0 and σ = 1), what is the value of IQR?
Hint - Solve for P75 – P25.
9b. If X is distributed normal (mean=μ and standard deviation = σ), what is the value of IQR in standard deviation units? Hint - Solve for the blank in the following expression. IQR = _____ σ2
9c. Using your answer to part “b”, what is solution for σ as a function of IQR?
Hint - Solve for the blank in the following expression. σ2 = _____IQR
10. SAT scores are distributed Normal with mean μ=500 and standard deviation σ=100, whereas IQ scores are distributed Normal with mean μ=100 and standard deviation σ=15. If it can be assumed that the SAT test is equivalent to an IQ test, what would a score of 720 on the SAT test have been on the IQ test?
11. Suppose it is known that, among persons 17 years of age and older, half the males and one-third of the females are smokers. A random sample of 10 males and 15 females is obtained.
11a. What is the probability that none are smokers?
11b. What is the probability that the sample contains 4 male smokers and 6 female smokers?