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1. As a part of a recent study investigating the effectiveness of the drug mifepristone for terminating early pregnancy, 488 women were administered the drug, followed 48 hours later by a single dose of a second drug, misoprostol. In 473 of these women, the pregnancy was terminated. a) Estimate the proportion of successfully terminated early pregnancies among women using the described regimen. b) Construct a 95% confidence interval for the true population proportion p. c) Interpret the confidence interval. d) Calculate a 90% confidence interval for p. e) How does the 90% confidence interval compare to the 95% interval?
2. Suppose that you are interested in determining whether the advice given by a physician during a routine physical examination is effective in encouraging patients to stop smoking. In a study of current smokers, one group of patients was given a brief talk about the hazards of smoking and was encouraged to quit. A second group received no advice pertaining to smoking. All patients were given a follow-up exam. In the sample of 114 patients who received the advice, 11 reported they had quit smoking. In the sample of 96 patients who had not, 7 had quit smoking. a) Estimate the difference in population proportions p1-p2. b) Construct a 95% confidence interval for this difference. c) At the 0.05 level of significance, test the null hypothesis that the proportions of patients who quit smoking are identical for those who received advice and those who did not. d) Do you believe that the advice given by physicians is effective? Why or why not?
3. Suppose you are interested in investigating the factors that affect the prevalence of tuberculosis among intravenous drug users. In a group of 97 individuals who admit to sharing needles, 24.7% had a positive tuberculin skin test result. Among 161 drug users who deny sharing needles, 17.4% had a positive test result. a) Assuming that the population proportions of positive skin test results are in fact equal, estimate their common value p. b) At the 0.05 level of significance, test the null hypothesis that the proportions of intravenous drug users who have a positive tuberculin skin test result are identical for those who share needles and those who do not. c) What do you conclude? d) Construct a 95% confidence interval for the true difference in proportions.
4. The dataset lowbwt contains information for a sample of 100 low birth weight infants born in two teaching hospitals in Boston, Massachusetts. Indicators of a maternal diagnosis of toxemia during pregnancy – a condition characterized by high blood pressure and other potentially serious complications – are saved under the variable name tox. The value 1 represents a diagnosis of toxemia and 0 means no such diagnosis. a) Estimate the proportion of low birth weight infants whose mothers experienced toxemia during pregnancy. b) Construct a 95% confidence interval for the true population proportion p.
In answering the questions below, be sure to specify the type of analysis you are referring to (e.g. margin of error, one-sample t-test, two-sample t-test, comparison of proportions, etc.), and why. Indicate whether you are using a one-tailed or a two-tailed test, and why. (This is because there are probably several legitimate ways to interpret the question, so we want to be sure that your answer matches the way you interpreted the question.) State the null hypothesis, and the specific alternative hypothesis used for the sample size/power calculation. (You’ll find it much easier to solve the problems if you write down clearly all of the information you have.) Don’t expect the answers to the questions to be “reasonable”.
5. Suppose you are evaluating a grant proposal. The proposer seeks $250,000 to conduct a series of experiments on patients with a rare kind of brain lesion. Let’s say the proposer will measure ability to remember lists of words in the experiment. A very interesting and provocative theory, described clearly in the proposal, predicts that the brain-damaged people should actually remember slightly better than the control group, and so the investigator proposes an experiment to test this prediction. A pilot study shows that among normals, the mean number of words remembered (from a list of 20) is 12.2, and the standard deviation is 3.1. (The SD in the lesion group could be assumed to be bigger, but we’ll just assume that it’s the same.) The investigator explains that a pool of 24 subjects with brain lesions is already available from a local clinic for the braindamaged group. Suppose that you, as a reviewer of the proposal, want a power of .80 in proposed experiments.
What effect size would be required to generate this power? (Assume a Type I error rate, i.e. alpha, of 0.05.)
Suppose that the effect size is actually 0.2 (small, as predicted by the proposed theory). What sample size would be needed to achieve a power of .80? (Should you fund the proposal?)
6. The body mass index is calculated by dividing a person’s weight by the square of his or her height; it is a measure of the extent to which the individual is overweight. A researcher would like to test the hypothesis that men who develop diabetes have a higher BMI than men of similar age who do not. A literature review indicates that in healthy men, BMI is normally distributed, with a mean of 25 and a standard deviation of 2.7. The researcher proposes to measure 25 normal and 25 diabetic men. It is felt that a difference in average BMI of 2.7 (that is, one standard deviation) would be clinically meaningful. What is the power of the proposed study?