Stat1600 Solution to Midterm #2 Form A

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1 Stat1600 Solution to Midterm #2 Form A 1. (10 points) According to the Centers for Disease Control and Prevention, 33.5% U.S. adults have high LDL, or bad, cholestrol. Given a random sample of n=12 U.S. adults, what is the probability that exactly 2 adults have high LDL. P (X= 2) = 12! 2!(12 2)! (.335)2 (1.335) 12 2 = ! (.335) 2 (.665) ! = (.335)2 (.665) 10 = 66 (.335) 2 (.665) 10 = According to a recent population survey, 24.7% of the State of Michigan population 25 years old and over have completed a bachelor s degree. Given a random sample of 100 persons in Michigan 25 years old or over, (a) (10 points) the number of persons who have completed a bachelor s degree is expected to be around, give or take or so. µ = np = = 24.7; σ = npq = 24.7 (1.247) = = The number of people who have completed a bachelor s degree is expected to be around 24.7, give or take (b) (10 points) what is the probability that out of the sample of n=100 that 27 or more have completed a bachelors degree? (Hint: Use the normal approximation of the binomial.) 1

2 P (X 27) = P (X > 26.5) ( ) P Z > = P (Z > 0.42) P (Z 0.42) = = = = = z z 3. According to Larsen, R. J., and Marx, M. L. (1986). An Introduction to Mathematical Statistics and Its Applications. Second Edition. Prentice-Hall, Englewood Cliffs, New Jersey, page 295: There is a theory that the anticipation of a birthday can prolong a person s life. In a study set up to examine that notion statistically, it was found that only 60 of 747 people whose obituaries were published in Salt Lake City in 1975 died in the three-month period preceding their birthday. (a) (10 points) What percentage of people in Salt Lake City in 1975 died in the threemonth period preceding their birthday? ˆp = = 0.08 i.e., 8%. (b) (10 points) Calculate the standard error for your estimate in (a) (1 0.08) SE = = i.e., 1.0%. 747 (c) (10 points) Calculate a 95% confidence interval for the true percentage. 2

3 ME = 1.96SE = = A 95% confidence interval for the true proportion is ( , ) = (0.060, 0.100). Hence a 95% confidence interval for the true percentage is (6.0%, 10.0%). 4. The table below shows the result of a study of the effectiveness of thalidomide in healing mouth ulcers in AIDS patients (an announcement by the National Institutes of Health on October 31, 1995): ulcers healed? YES NO Treatment 14 9 Placebo 1 21 (a) (10 points) Estimate the difference in heal rates (i.e., difference in proportions) of the two groups (Treatment versus Placebo). Let group 1 = Treatment and group 2 = Placebo. Then n 1 = = 23; n 2 = = 22. It follows ˆp 1 = = 0.609; ˆp 2 = 1 22 = The difference in proportions is estimated by ˆp 1 ˆp 2 = = (b) (10 points) Calculate a 95% confidence interval for the difference in heal rates. Does the interval exclude 0? Is the difference statistically significant? ANSWER (show your work): 3

4 From part (a), we can calculate the standard error of the estimated difference in proportions:.609 (1.609).045 (1.045) SE = = = =.111 Hence the margin of error is ME = =.218. Consequently, we can get a 95% c.i. for p 1 p 2 : ( , ) = (0.346, 0.782) which excludes 0. Therefore, the proportions differ and it appears that the heal rate for the Treatment group is greater than that of the Placebo group. Note that the interpretation of this 95% confidence interval: we are 95% confident that the difference in proportions (percentages) is in between (34.6%) and and (78.2%). ulcers healed? YES NO Treatment 14 9 Placebo 1 21 (c) (10 points) Estimate the odds ratio of those who were healed (YES) in the two groups. Interpret your answer in words. The odds ratio is estimated as OR = odds 1 odds 2 = 14/9 1/21 = That is, the odds for those in the Treatment group to be healed is 32.7 times as high as that for those in the Placebo group. (d) (10 points) Estimate the ratio of proportions of those who were healed (YES) in the two groups. Interpret your answer in words. 4

5 The ratio of heal rates is estimated as RR = ˆp 1 ˆp 2 = = 13.5 That is, those in the Treatment group are 13.5 times more likely to be healed than those in the Placebo group. 5. EXTRA CREDIT PROBLEM. (5 points) In the past, numerous epidemiological studies showed that women who were taking combined Hormone Replacement Therapy (HRT) also had a lower-than-average incidence of Coronary Heart Disease (CHD), leading doctors to propose that HRT was protective against CHD. Certainly, the claim was false. List at least two possible confounders and pick one of them to show the pathway graph that connects the confounder, HRT, and CHD. (Source: Wikipedia) There exist possible confounders that could be probable causes to the lower-thanaverage incidence of CHD. For instance, higher socio-economic status and better nutrition. A possible pathway could be higher socio-economic status = lower CHD taking HRT 5