STAT101A Biostatistics Problem Set 2 29 September 2020 Instruction: Solve the problems specified below. You can use statistical software of your choice to aide you with the problem solving. 1. Cancer, Epidemiology: An experiment is designed to test the potency of a drug on 20 rats. Previous animal studies have shown that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. a. What is the probability that 3 or more rats will die in the first 4 hours? b. Suppose 2 rats die in the first 4 hours. What is the probability that 2 or fewer rats will die in the next 4 hours? c. What is the probability that 0 rats will die in the 8-hour period? d. What is the probability that 1 rat will die in the 8-hour period? e. What is the probability that 2 rats will die in the 8-hour period? f. Can you write a general formula for the probability that x rats will die in the 8 -hour period? Evaluate this formula for x = 0, 1, . . . , 10. (Hint: You can use the BINOMDIST function of Microsoft Excel, and dbinom or pbinom of R) 2. Hospital Epidemiology: Suppose the number of admissions to the emergency room at a small hospital follows a Poisson distribution, but the incidence rate changes on different days of the week. On a weekday there are on average two admissions per day, while on a weekend day there is on average one admission per day. a. What is the probability of at least one admission on a Wednesday? b. What is the probability of at least one admission on a Saturday? c. What is the probability of having 0, 1, and 2+ admissions for an entire week, if the results for different days during the week are assumed to be independent? 3. Environmental Health: Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. a. The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion? b. The hospital also finds that on high-pollution days the number of admissions is Poisson-distributed with mean = 4.0 admissions per day. c. Answer (a) for high-pollution days. d. On a random day during the year, what is the probability there will be 4 admissions to the emergency ward, assuming there are 345 normal-pollution days and 20 high-pollution days? e. Answer (a) for a random day during the year. 4. Veterinary Medicine: Suppose 6 of 15 pigs in farm develop swine flu, whereas 20% of pigs nationwide develop swine flu. Is there evidence of an excessive number of cases in the group of pigs? That is, what is the probability of obtaining at least 6 cases in this class if the nationwide rate holds true? a. What is the expected number of students in the class who will develop swine flu? b. What is the probability of obtaining exactly 6 events for a Poisson distribution with parameter μ = 4.0? c. What is the probability of obtaining at least 6 events for a Poisson distribution with parameter μ = 4.0? d. What is the expected value and variance for a Poisson distribution with parameter μ = 4.0? STAT101A Biostatistics Problem Set 2 29 September 2020 5. Endocrinology: Consider the Data Set BONEDEN.DAT on the Companion Website. Calculate the difference in bone density of the lumbar spine (g/cm2) between the heavier-smoking twin and the lightersmoking twin (bone density for the heavier smoking twin minus bone density for the lighter-smoking twin) for each of the 41 twin pairs. Suppose smoking has no relationship to bone density. a. What would be the expected number of twin pairs with negative difference scores? What is the actual number of twin pairs with negative difference scores? Do you feel smoking is related to bone density of the lumbar spine, given the observed results? Why or why not? A yes/no answer is not acceptable. (Hint: Use the binomial distribution.) b. Sort the differences in smoking between members of a twin pair (expressed in pack-years). Identify the subgroup of 20 twin pairs with the largest differences in smoking. c. Answer Problem (a) based on this subgroup of 20 twin pairs. d. Answer Problem (a) for bone density of the femoral neck. e. Answer Problem (b) for bone density of the femoral neck. f. Answer Problem (c) for bone density of the femoral shaft. g. Answer Problem (b) for bone density of the femoral shaft. 6. Veterinary Dentistry: The data in Table 4.19 were reported by veterinarians in the Health Professionals Follow-up Study on the number of teeth lost over a 1-year period (January 1, 1987 to December 31, 1987). Please see the table provided below. a. If we assume the average number of teeth lost in the 5−9 group is 7 teeth and the average number of teeth lost in the 10+ group is 12 teeth, what is the best estimate of the average number of teeth lost per year? b. Suppose that on January 1, 1987, a dog is 5 years old, that it will live for 3 more years (until 2016), and that the rate of tooth loss over this 3-year period is the same as in 1987. If a dog has 13 teeth remaining on January 1, 1987, what is the probability it will need dentures (have 10 or fewer teeth remaining) during its 3-year lifetime? (Hint: Use the Poisson distribution.) c. Suppose dental practice improves over the 3-year period. We assume the rate of tooth loss per year from 1987−2001 (15 years) is the same as in 1987, whereas the rate of tooth loss per year fr om 2002−2016 (15 years) is half the 1987 rate. What is the probability that the dog in Problem (b) will require dentures under these altered assumptions? (Hint: Consider a mixture of two Poisson distributions.)
