You have data on 100 defendants who have been on trial twice, and the verdict in
ID: 3274523 • Letter: Y
Question
You have data on 100 defendants who have been on trial twice, and the verdict in each trial. Of those: 20 were never convicted, 40 were convicted both times, 10 were convicted the first time but not the second, and 30 were convicted the second time but not the first. What is the probability a defendant randomly selected from among those convicted in their first trial was convicted in their second trial? Is the outcome of the second trial independent of the outcome of the first? Explain. Suppose someone develops a screening for a particular learning problem. A school district proposes conducting the screening on their entire first grade population, since interventions to correct the problem are more effective when delivered carlier. The screening produces a false negative in 1 in 25 with the condition and a false positive in 1 in 50 without the condition. One in 200 students has the condition. What is the probability a student with a negative result has the condition? What is the probability a student with a positive result has the condition? Based on these two answers, do you see any problem? If so, can you think of a reasonable solution? If so, what? After implementing your solution, what is the probability you will have incorrectly concluded a student has the problem when they in fact do not? You think the probability the Gators will beat the Seminoles (in football) is 1/2 and the probability the Gators will beat the Bulldogs is 1/3. You think these are independent. You have a wager with a friend in which you will win a case of beer if the Gators beat the Seminoles and lose to the Bulldogs, two cases if the Gators beat the Bulldogs and lose to the Seminoles, and six cases if the Gators win both. If the Gators lose both, you owe her three cases. Write out the probability distribution and calculate the expected value and standard deviation of your winnings. How can both you and your friend think this bet is a good deal? A factory has 22 identical machines. The expected number of break downs for each machine is 1.8 per year, with a standard deviation of 1.2. What are the mean and standard deviation of the total number of breakdowns each year? If each repair costs $1,000, what are the mean and standard deviation of annual repair costs? The population mean on a statistics exam is 72, with a standard deviation of 12. The population average on the class project is 95, with a standard deviation of 4. If the exam is 70% of the final grade and the project 30%, what are the mean and standard deviation of final grades? The population mean score for a particular exam is 74 and the standard deviation is 9. What are the mean and standard deviation of the class average score for classes composed of 36 students randomly drawn from the population?Explanation / Answer
1. Let A denote convicted the first time, B denote convicted the second time
P(Ac and Bc) = 0.20
P(A and B) = 0.40
P(A and Bc) = 0.10 and P(Ac and B) = 0.30
1.1. P(Ac and B) = 0.30
P(B) - P(A and B) = 0.30
P(B) = 0.30 + P(A and B)
= 0.3 + 0.40 = 0.70
The probability a defendant randomly selected from among those convicted in their first tras was convicted in their second trial is
P(A/B) = P(A and B) / P(B) = 0.40 / 0.70 = 0.5714
1.2) P(A and Bc) = 0.10
P(A) - P(A and B) = 0.10
P(A) = 0.10 + P(A and B)
= 0.10 + 0.40 = 0.50
Now
P(A/B)=P(A and B) / P(B) = 0.40/0.70 = 0.5714 which is not equal to P(A)
P(B/A) = P(A and B) / P(A) = 0.40 / 0.50 = 0.8 which is not equal to P(B)
Therefore, A and B are not independent
2. From the given data
2.1 The Probability a student with a negive result has the condition is 1/75
2.2. The Probability a student with a positive result has the condition is 24/75
2.3 The probability you will have incorrectly concluded a student has the problem when they in fact do not is 49/75
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