Name: ___________________________________Bayesian Statistics, 22S:138Midterm 1, Fall 2006PRACTICE PROBLEMS for Midterm 1, Fall 2007Show any computations that you carry out. Use the back of your exam paper if yourun out of space. Point values for each question are shown in parentheses.1. The following facts described the students who took 22S:138 in a recent year:• 35% of the students were statistics grad students• 25% of the students were biostatistics grad students• 40% of the students were in neither of the above categories• 60% of the statistics students were women• 75% of the biostatistics students were women• 40% of the students in other categories were women(a) You drew a student at random from th e class list for 22S:138 in that year.What is the probability that the student you drew is a woman?(b) Suppose that the student you drew was a woman. What is the probabil-ity that she was not a statistics grad student and not a biostatistics gradstudent, given that she was a woman?12. A social worker did a Bayesian analysis to estimate the proportion p of patientsdischarged from a community hospital who were satisfied with the care they re-ceived while in the hospital.(a) Before collecting any new data on the subject, she studied in formation oncomplaints filed with the hospital and talked with nurses, chaplains, andothers who interacted with patients in order to get a rough idea about p.Based on th e preliminary assessment, she expressed her knowledge about pin the form of a Beta prior:p ∼ Beta(4, 6)i. What is the mean of the prior distribution? (numeric answer)(b) The social worker then chose a simple random sample of 30 patients who hadbeen discharged from the hospital during the last 12 months. She interviewedeach one, and asked whether they were s atisfied with the care they received.18 patients said “yes” and 12 said “no.”i. What parametric family is appropriate for the likelihood for these data?(just the name)(c) The social worker combined her prior with the likelihood and obtained thefollowing posterior distribution:p(p|y) = Beta(22, 18)She used R to calculate a 95% equal-tail credible set for p, with the followingresults:> qbeta(c(0.025, 0.975), 22,18)[1] 0.3962080 0.6990522i. Circle all the statements below that are true regarding this Bayesianinterval.A. 95% of hospital patients have a satisfaction score between 0.396 and0.699B. For a person who agreed with the social worker’s prior, the prob-ability is .95 th at the population proportion p lies in the interval(0.396,0.699).C. 95% of s imple random samples will produce a sample proportion inthe interval (0.,396,0.699)D. If we took many, many simple random samples of size 30 from thepopulation of patients discharged from th is hospital, 95% of the sam-ples would produce a Bayesian interval that contained the true p2E. none of the aboveii. The social worker decides that she would like to estimate p more precisely— that is, she wants a narrower credible set. So she plans to draw a newsimple random s ample of 10 patients. She wants to calculate an 80%prediction interval for the number of “yeses” that she will get in the newsample. She uses the pbetap function, with the following output:> pbetap( 22,18,10,0:10)s pred[1,] 0 0.001026584[2,] 1 0.008364758[3,] 2 0.033298173[4,] 3 0.085243322[5,] 4 0.155391473[6,] 5 0.210791911[7,] 6 0.215582637[8,] 7 0.164253437[9,] 8 0.089312807[10,] 9 0.031337827[11,] 10 0.005397070A. What is the probability that she will get 3 “yeses” in her new sampleof 10 people? (numeric answer)B. What are the endpoints of the 80% prediction interval? (numericanswer; show your work)33. The data value y is a random draw from a probability distribution depending ona parameter R:p(y|R) ∝ R1/2exp −R (y − 6)22!−∞ < y < ∞R > 0(a) (3) What parametric family is this? (Name and parameter(s)).(b) (3) To what parametric family of distributions would the conjugate prior forR belong? (Ju st name the family.)4. Researchers choose a simple random sample of counties from amongst all thecounties in the U.S. The variable rep orted for each of the selected counties is thearea of the county in square miles.(a) Which of the following parametric families would b e most likely to be appro-priate for the likelihood in their analysis?i. betaii. binomialiii. gammaiv. normalv. poisson(b) Justify your answer in a sentence or two.45. Two fair dice, one red and one green, are rolled. Let the event A be “the sumof the faces showing is an even number.” Let event B be “the sum of the facesshowing is divisible by 3.”(a) List the items in A, and find P (A).(b) List the outcomes in B, and find P (B).(c) List the outcomes in ATB and find P (ATB).(d) Are the events A and B independent? Briefly explain why or why
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