Homework 2 (due on Monday, February 6)

• Problems will be added until the last class before the due date.
• No electronic submission unless explicitly allowed.
• Use your own words when answering questions. Copying from other sources (including textbooks, handouts, my blog) is strongly discouraged, because it often indicates you don't understand your answer.

1. Because serum cholesterol is related to age and sex, some investigators prefer to express it in terms of z-scores. If X is raw serum cholesterol, then z = (X - μ)/σ, where μ is the mean and σ is the standard deviation of serum cholesterol for a given age-sex group. Suppose z is regarded as a standard normal distribution. Also suppose a person is regarded as having high cholesterol if z > 2.0 and borderline cholesterol if 1.5 < z < 2.0. [Make sure you can get the answers from both tables and Stata, although you don't need to turn in Stata output.]
   • What is Pr(z < 0.5)?
   • What is Pr(z > 0.5)?
   • What is Pr(-1.0 < z < 1.5)?
   • What proportion of people have high cholesterol?
   • What proportion of people have borderline cholesterol?
2. Beta-carotene is hypothesized to prevent macular degeneration (an important eye disease in the elderly). A dietary survey was undertaken to measure beta-carotene intake in the typical American diet. Assume the distribution of ln(beta-carotene intake) is normal, with mean 8.34 and standard deviation 1.00. [Units are in ln(IU). ln = natural logarithm.]
   • What percentage of people have dietary beta-carotene intake below 2000 IU? (Note that ln(2000)=7.60.)
   • What percentage of people have dietary beta-carotene intake below 1000 IU? (Note that ln(1000)=6.91.)
   • Some studies suggest beta-carotene intake over 10,000 IU may protect against macular degeneration. What percentage of people have a dietary intake of at least 10,000 IU?
   • Suppose each person took a beta-carotene supplement pill of dosage 5000 IU in addition to his or her normal diet. Assume the resulting distribution of ln(beta-carotene intake) is normally distributed, with mean 9.12 and standard deviation 1.00. What percentage of people would have an intake from diet and supplements of at least 10,000 IU?
3. Much discussion has appeared in the medical literature in recent years on the role of diet in the development of heart disease. The serum-cholesterol levels of a group of people who eat a primarily macrobiotic diet are measured. Among 24 of them, aged 20-39, the mean cholesterol level was found to be 175 mg/dL with a standard deviation of 35 mg/dL.
   • If the mean cholesterol level in the general population in this age group is 230 mg/dL and the distribution is assumed to be normal, then test the hypothesis that the group of people on a macrobiotic diet have cholesterol levels different from those of the general population.
   • Compute a 95% confidence interval for the true mean cholesterol level in this group.
   • What type of complementary information is provided by the hypothesis test and confidence interval in this case?
4. One method for assessing the bioavailability of a drug is to note its concentration in blood and/or urine samples at certain periods of time after giving the drug. Suppose we want to compare the concentrations of two types of aspirin (types A and B) in urine specimens taken from the same person, 1 hour after he or she has taken the drug. Hence, a specific dosage of either type A or type B aspirin is given at one time and the 1-hour urine concentration is measured. One week later, after the first aspirin has presumably been cleared from the system, the same dosage of the other aspirin is given to the same person and the 1-hour urine concentration is noted. Because the order of giving the drugs may affect the results, a table of random numbers is used to decide which of the two types of aspirin to give first. This experiment is performed on 10 people; the results are given in this table. Suppose we want to test the hypothesis that the mean concentrations of the two drugs are the same in urine specimens.
   • What are the appropriate hypotheses?
   • What are the appropriate procedures to test these hypotheses? Conduct the tests mentioned.
   • What is the best single-number estimate of the mean difference in concentrations between the two drugs? (A single-number estimate is also called a point estimate.)
   • What is a 95% CI for the mean difference? (A CI is also called an interval estimate.)
   • Suppose a significance level of .05 is used in your test. What is the relationship between the decision reached with the test procedure and the nature of the confidence interval?
5. True or false? If false, state (1) why it is wrong and (2) the correct results/statements/conclusions that is beyond just grammatically negating a false statement.
   • In a test to compare the birth weights of children born to 14 smokers with those of children born to 15 non-smokers, the p-value was 0.0064. So the probability of the null hypothesis of no difference is 0.0064.
   • In a clinical trial to compare the effect of a drug with that of placebo, both the drug and placebo effects were measured for every subject (assuming carry-over effect is ignorable). We can calculate the average of the drug effects, the average of the placebo effects, and the average of per-subject drug-placebo effect differences. Since the difference of the first two averages is the same as the third average, we can carry out either a two-sample t-test or a paired t-test to give us the same results.
   • For a sample, we want to know its variation or how the data spread out. There are two ways of capturing variation: standard deviation and inter-quartile range. The two measures are similar numerically and are effectively interchangeable.