4. The CDC estimated that in 2008 there were 110 million individuals with new and exist- ing sexually transmitted infections (STIS) in the U.S. The CDC report included 8 com- mon STIS: chlamydia, gonorrhea, hepatitis B virus, herpes simplex virus type 2, human immunodeficiency virus (HIV), human papillomavirus (HPV), syphilis, and trichomo- niasis. [Note: If an individual has multiple STIs, say, 3, the individual is counted as 1, not 3.] In 2008 the U.S. population was estimated to be 304.1 million. Suppose a simple random sample of size 1000 was chosen, without replacement, from the entire U.S. population of 304.1 million individuals. (a) (8 points) Let r.v. X be the number of people with at least 1 of the 8 STIs in the sample. X has a hypergeometric distribution. What are the possible values of X? What are the values of the parameters of the distribution of X? Write the exact distribution of x below. 3 P[X = x] = What is the shape of the distribution? How do you know the shape without graph- ing the distribution? (b) (6 points) You know that the binomial distribution has two parameters: n and p. If you resort to using the binomial distribution (instead of the hypergeometric dis- tribution) to find the probability that the number of infected individuals in the sample is at least 300 but at most 360, what are the values of n and p? Find the actual probability that the number of infected individuals in the sample is at least 300 but at most 360. To receive full credit, first write the probability expression you wish to evaluate and then evaluate the expression. (c) (3 points) The binomial distribution requires that sampling is done with replace- ment to ensure that the probability of success remains the same from trial to trial. Why can we use the binomial distribution to solve the preceding problem? 4. The CDC estimated that in 2008 there were 110 million individuals with new and exist- ing sexually transmitted infections (STIS) in the U.S. The CDC report included 8 com- mon STIS: chlamydia, gonorrhea, hepatitis B virus, herpes simplex virus type 2, human immunodeficiency virus (HIV), human papillomavirus (HPV), syphilis, and trichomo- niasis. [Note: If an individual has multiple STIs, say, 3, the individual is counted as 1, not 3.] In 2008 the U.S. population was estimated to be 304.1 million. Suppose a simple random sample of size 1000 was chosen, without replacement, from the entire U.S. population of 304.1 million individuals. (a) (8 points) Let r.v. X be the number of people with at least 1 of the 8 STIs in the sample. X has a hypergeometric distribution. What are the possible values of X? What are the values of the parameters of the distribution of X? Write the exact distribution of x below. 3 P[X = x] = What is the shape of the distribution? How do you know the shape without graph- ing the distribution? (b) (6 points) You know that the binomial distribution has two parameters: n and p. If you resort to using the binomial distribution (instead of the hypergeometric dis- tribution) to find the probability that the number of infected individuals in the sample is at least 300 but at most 360, what are the values of n and p? Find the actual probability that the number of infected individuals in the sample is at least 300 but at most 360. To receive full credit, first write the probability expression you wish to evaluate and then evaluate the expression. (c) (3 points) The binomial distribution requires that sampling is done with replace- ment to ensure that the probability of success remains the same from trial to trial. Why can we use the binomial distribution to solve the preceding
