STAT 1012: Statistics for Life Sciences 2015/16 Term 2 Assignment #2 Due: March 7th, 2016 (Monday) at 6:30pm Total Score: 100 points This assignment covers material from Chapter 2 to 3 of the lecture notes. Please present your answers in 4 significant figures. Note that the solutions will be available on March 8th (Tuesday) at 5pm, as the Mid-term exam will be on March 14th. No late assignment will be accepted after the solutions are posted. Problem 1 [21 points]: Consider the prevalence (i.e. proportion) of Alzheimer's disease (, or commonly called ) by sex and age group displayed in Table 1: (a) [8 points] Suppose that 3 persons (male age 78, male age 83 and female age 73) are selected from a community. Given that two of them have Alzheimer's disease, what is the conditional probability that both of them are below age 80? (b) [7 points] Suppose that we are only interested in couples with ages both in the range of 70-74, and the probability of at least one of the couples having Alzheimer's disease is 2.48%,what is the probability that both of them have Alzheimer's disease? Do you think that the presence of Alzheimer's disease for wife is independent to the presence of Alzheimer's disease for husband? (c) [6 points] Table 2 shows the age-sex distribution of people in a retirement community. What is the prevalence of Alzheimer's disease of the community? Problem 2[16 points]: Suppose screening test A has been used to detect disease D. Based on historical performance, it's known that 20% of patients who are A positive will have disease D, while 98% of patients who are A negative will not have disease D. A new screening test (Test B) is now purposed, and we would like to verify if it performs better than Test A. It's known that 80% of patients who are B positive will have disease D. It is known that 10% of the population are detected to be A positive, while 1% of the population are detected to be B positive. (a) [3 points] Show that the rate of disease D in the population is 3.80%. (b) [5 points] Based on the result in part (a), what is the probability of disease D given B negative? (c) [8 points] Compute the relative risk of disease D given A positive, and the relative risk of disease D given B positive. What conclusion can you draw? Page 1/2 Problem 3 [12 points]: Mary doesn't know the blood type of her father, but she knows that her mother is of blood type B (i.e. either genotype BO or BB). Given that the genotype of Mary's grandparents (from the father's side) are both BO, and the ratio of genotype BO and BB in the population is 14:3. What is the probability that Mary is of blood type B? Problem 4[18 points]: In a study of HIV infection among intravenous drug users, 20% of light users (who had 100 injections per month) were HIV positive, while 45% of heavy users (who had > 100 injections per month) were HIV positive. (a) [4 points] Let X be the # of HIV positive out of 5 light users. What is the probability mass function of X? (b) [3 points] Sketch the cumulative distribution function of X. (c) [6 points] Suppose that a group of 5 light users and 2 heavy users were selected. What is the probability of exactly one HIV positive? (d) [5points] Given that exactly 1 HIV positive is found from a group 3 people (2 light users and 1 heavy user), how likely the HIV positive is a heavy user? Problem 5 [16 points]: During a normal-pollution day, Let X be the number of admissions to the emergency room of a hospital, which follows Poisson distribution with mean=0.75 admissions per day. (a) [5 points] What is the probability mass function Pr(X=x) for x = 0, 1, 2 and 3? (b) [4 points] If the hospital want to accept all the admissions during a normal-pollution day with at least 90% probability, show that a minimum of 2 beds are required in the room. (c) [7 points]During a high-pollution day, the number of admissions follow Poisson distribution with mean=1.5 admissions per day. If the hospital wants to accept all the admissions with at least 90% chance, what is the minimum number of beds required? Problem 6 [17 points]: In carcinogenesis, it is assumed that a malignant () cell is mutated () from an intermediate () cell, and an intermediate cell is mutated from a normal cell. In order for a cancer to develop, there should be at least one malignant cell in the body. The model has been applied to the development of breast cancer in females. Suppose there are 108 normal breast cells and 0 intermediate and malignant breast cells among females of age 20. Assume that a cell is not able to undergo two mutations within a year, and the probability of a normal breast cell to mutate to an intermediate cell is 3.810-8 per year. (a) [3 points] What is the expected number of intermediate breast cells by age 21? (b) [7 points] Based on Poisson approximation to Binomial, approximate the probability of at least 2 intermediate cells by age 21. Do you think the approximation is good? (c) [7 points] Suppose the probability of an intermediate cell to mutate to a malignant cell is 5x10-6 per year, and a woman has 400 intermediate cells at age 50. Based on Poisson approximation to Binomial, what is the probability of having breast cancer by age 51? - End of Assignment Page 2/2