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8. Read problem 3.15 (pages 67-68) and then answer the questions below. a. What proportion in the sample had a culture test? What proportion had the Strep A OIA test? What are the probabilities of outcomes in the sample for those with the Culture test? What are the probabilities of outcomes in the sample for those with the Strep A OIA test? Make a tree to illustrate the probabilities in a-b-c. b. c. d.Use numbers in the tree to express the probability that a subject in the study really had a true positive status? That is, use law of total probability for this calculation. This can be calculated from the numbers in the table as well, but use the probabilities in a-d to find it. e. f. Usenumbers in a-e to compute the probability that a subject had a culture done given that he/she is a true positive? That is, use Bayes rule. This can be calculated from the numbers in the table as well, but use the probabilities in a-d to find it.nuaeuring plant. S be selected randomly for inspection. Find the probabilities of the following events. a uppose that in a tray containing 20 wafers 4 are defective. Two wafers are to Neither is defective. b At least one of the two is defective 3.12 In the setting of Exercise 3.11, answer the same three questions, assuming this time that only 2 a 3.13 The frequency of twin births has been increasing in recent years so that in 2000 about 3% of all 3.14 By using the definition of conditional probability, show that. c Neither is defective, given that at least one is not defective. the 20 wafers are defective. births were twins. Twins may be either identical or fraternal. The probability of a birth being that of mong identical twins has remained fairly constant at 0.4%. What proportion of twin births is identical? P(ABC) P(A)P(BIA)P(CIAB). 3.15 Children and adults with sore throats are often tested for strep throat. If untreated, strep throat can lead to rheumatic fever. The traditional method for assessing whether or not someone has strep throat is a culture. Because the results of the culture take a day to obtain, more rapid tests are often used. The Biostat A Optical Immunoassay (Strep A OIA), first developed in the early 1990s, is one of the rapid tests. Heiter and Bourbeau (1995) conducted a study in which the results of both the cultureChopter 3 Conditional Probobility and Independence 68 and Strep A OIA were obtained for 801 patients who potentially had strep throat. The resuits the following table. he results are in Number of Results True Falseru Fase Positive Positive Negative Negative Test Culture 239 7 0 Strep A OIA 225 21 52629 Answer the following questions based on the information given in the preceding table. What is the sensitivity of the culture? b a What is the specificity of the culture? What is the predictive value of the culture? Using the results presented in the table in Exercise 3.15, answer the following questions. a What is the sensitivity of the Strep A OIA? b What is the specificity of the Strep A OIA? c What is the predictive value of the Strep A OLA? In a certain village, the eldest daughter is expected to tend the familys flock of sheep. Suppose that every family in this large village has two children. What proportion of all daughters are eldest daughters? What assumptions did you make? Recall the dice game played by the Chevalier de Méré described in Exercise 2.58. Suppose a player bets that at least one 6 would appear during a total of four rolls. However, after the first two rolls, neither of which resulted in a 6, she had to leave the game. What proportion of her bet should be returned to her for this to be a fair game? (A game is fair if the amn 3.16 3.17 3.18 prohahility of

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Answer #1

a) Given n = 801

Proportion in the sample had a culture test = 801/1602 = 1/2

Proportion in the sample had the Strep A OIA test = 801/1602 = 1/2

b)

The probabilities of outcomes in the sample for those with the Culture test are

The probabilities of outcome in the sample for true positive with the Culture test = 239/801 = 0.2983

The probabilities of outcome in the sample for false positive with the Culture test = 7/801 = 0.0087

The probabilities of outcome in the sample for true negative with the Culture test = 555/801 = 0.6928

The probabilities of outcome in the sample for false negative with the Culture test = 0/801 = 0

c)

The probabilities of outcomes in the sample for those with the Culture test are

The probabilities of outcome in the sample for true positive with the Strep A OIA test = 225/801 = 0.2808

The probabilities of outcome in the sample for false positive with the Strep A OIA test = 21/801 = 0.0262

The probabilities of outcome in the sample for true negative with the Strep A OIA test = 526/801 = 0.6566

The probabilities of outcome in the sample for false negative with the Strep A OIA test = 29/801 = 0.0362

d)

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