The Taxicab Problem
In the city of Chennai (formerly Madras), India, a taxicab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in Chennai. Although the two companies are roughly equal in size, 85 percent of all cab accidents in Chennai involve Green cabs, whereas only 15 percent involve Blue cabs. A witness to the accident identified the car as a Blue cab. The police tested the witness's ability to identify cabs at night (when the colors blue and green are hard to distinguish). When presented with a sample of cabs, half of which were blue and half of which were green, the witness made a correct identification in 80 percent of the cases and erred in 20 percent of the cases. What is the probability that the cab involved in this particular accident was Blue rather than Green?
Answer is 41 %
Forensic psychologist Dr. P. Gupta-Phaer wants to know whether ordinary people (as opposed to trained statisticians) are able to give unbiased estimates of the correct answer to the Taxicab Problem, without doing any formal computations. After all, ordinary people might reasonably be called upon to serve as jurors in criminal trials involving information of this sort. If they are able to estimate such quantities in an unbiased way, justice will be served. But if people are biased in their estimates, then there is a danger that the guilty may go free, or that innocent persons may be punished. [See the Technical Note at the end of this problem for a fuller discussion of what "bias" means in the context of this problem.]
Thus, Dr. Gupta-Phaer gives this problem to 18 subjects, asking each of them to estimate the probability that the taxicab involved in this particular accident was Blue rather than Green. They were asked to express their estimates on a 0-to-100 percent scale, where 0 means that the cab definitely was not Blue, and 100 means that it definitely was Blue. The subjects' responses are listed in the table below. Drawing on these data and on the problem description, do each of the following:
--------------------------------------…
Probability Estimates of 18 Subjects
--------------------------------------…
50 60 45 80 25 85
60 50 55 65 50 40
65 15 85 50 60 50
--------------------------------------…
(a) In symbols, state both the research hypothesis and the null hypothesis. As always, make sure your research hypothesis matches the problem with respect to directionality. [Hint: To construct these hypotheses you must first decide what the mean of a sample of unbiased estimates would be (be sure to read the Technical Note). This mean can be used as the value of μ0.
In the city of Chennai (formerly Madras), India, a taxicab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in Chennai. Although the two companies are roughly equal in size, 85 percent of all cab accidents in Chennai involve Green cabs, whereas only 15 percent involve Blue cabs. A witness to the accident identified the car as a Blue cab. The police tested the witness's ability to identify cabs at night (when the colors blue and green are hard to distinguish). When presented with a sample of cabs, half of which were blue and half of which were green, the witness made a correct identification in 80 percent of the cases and erred in 20 percent of the cases. What is the probability that the cab involved in this particular accident was Blue rather than Green?
Answer is 41 %
Forensic psychologist Dr. P. Gupta-Phaer wants to know whether ordinary people (as opposed to trained statisticians) are able to give unbiased estimates of the correct answer to the Taxicab Problem, without doing any formal computations. After all, ordinary people might reasonably be called upon to serve as jurors in criminal trials involving information of this sort. If they are able to estimate such quantities in an unbiased way, justice will be served. But if people are biased in their estimates, then there is a danger that the guilty may go free, or that innocent persons may be punished. [See the Technical Note at the end of this problem for a fuller discussion of what "bias" means in the context of this problem.]
Thus, Dr. Gupta-Phaer gives this problem to 18 subjects, asking each of them to estimate the probability that the taxicab involved in this particular accident was Blue rather than Green. They were asked to express their estimates on a 0-to-100 percent scale, where 0 means that the cab definitely was not Blue, and 100 means that it definitely was Blue. The subjects' responses are listed in the table below. Drawing on these data and on the problem description, do each of the following:
--------------------------------------…
Probability Estimates of 18 Subjects
--------------------------------------…
50 60 45 80 25 85
60 50 55 65 50 40
65 15 85 50 60 50
--------------------------------------…
(a) In symbols, state both the research hypothesis and the null hypothesis. As always, make sure your research hypothesis matches the problem with respect to directionality. [Hint: To construct these hypotheses you must first decide what the mean of a sample of unbiased estimates would be (be sure to read the Technical Note). This mean can be used as the value of μ0.