Problem Set and Review Questions 2

Transcription

1 Problem Set and Review Questions 2 1. (Review of what we did in class) Imagine a hypothetical college that offers only two social science majors: Economics and Sociology. The economics department has 4 male professors and 2 female professors. The sociology department has 1 male professor and 5 female professors. In the economics department, the professors earn the following salaries: Professor A (male): $95,000 Professor B (male): $90,000 Professor C (male): $90,000 Professor D (male): $100,000 Professor E (female): $105,000 Professor F (female): $110,000 In the sociology department, the professors earn the following salaries: Professor G (male): $60,000 Professor H (female): $60,000 Professor I (female): $65,000 Professor J (female): $70,000 Professor K (female): $70,000 Professor L (female): $65,000 a) Calculate the average salary of male professors in the social sciences and the average salary of female professors in the social sciences. Which gender has the higher average? b) Calculate the average salary of male professors in economics and the average salary of female professors in economics. Which gender has higher salaries in this field? c) Calculate the average salary of male professors in sociology and the average salary of female professors in sociology. Which gender higher has salaries in this field? d) How do you explain the combined results from a, b, and c? 2. (Review of what we did in class) Two basketball teams played against each other twice. Alex is Team A s best three-point shooter, and Bob is Team B s best three-point shooter. In game 1, Alex had 6 hits out of 16 three-point shots, and in game 2 he had 1 hit out of 5 attempts. Bob, on the other hand, scored 2 out of 5 shots in game 1, and 3 out of 12 shots in game 2. a) Who was the best 3-point shooter in game 1? b) Who was the best 3-point shooter in game 2? c) Who was the best 3-point shooter overall?

2 d) How do you explain the results? 3. Imagine that a population is divided into two groups, A and B, and that we calculate the average IQ of group A and the average IQ of group B. Suppose now that some members of group A leave the group and join group B. Given this information, describe whether the following results are possible or not: a) The average IQ of one group increases and the average IQ of the other group decreases; b) The average IQ of both groups increases; c) The average IQ of both groups decreases. 4. If there is random migration from one group to another, what determines the probability of observing the Will Rogers phenomenon? Explain. 5. Suppose Group A has n members and Group B has m members. If a person from Group A moves to the Group B, what is the lowest possible probability that we will observe the Will Rogers phenomenon? What is the highest probability? 6. If the median is used as the measure of central tendency instead of the average, is it still possible to observe the Will Rogers phenomenon? Explain. 7. What is the most important implication of the Will Rogers phenomenon? 8. Consider the following three hypothetical statements: a) Cigarettes were not harmful to my grandfather. He smoked since he was 14 and he lived to be 90 years old. b) The stock market did not react well to the recent increase in the Federal Funds Rate yesterday. The Dow Jones fell 3% and NASDAQ fell 4% today. c) The death penalty does not deter murder. Texas has the highest rate of executions but the murder rate there is equal to the US average. These statements have two main problems: the first one, which will be discussed in more depth later during the course, is that they are using a sample size of one to reach their conclusions. But there is a second problem, which is more basic. Explain what the second problem is. 9. (Review of what we did in class) Imagine two drivers. Driver A drives a gas-guzzler that gets only 8 miles/gallon, whereas Driver B drives a midsize sedan that gets 30 miles/gallon. Both drivers are thinking about getting a car that is more efficient than the ones they have right now in order to save money on gas. Driver A is considering a car that gets 10 miles/gallon, whereas Driver B is thinking about trading his sedan for the most efficient car on the market, which gets

3 50 miles/gallon. Call X the distance in miles that these two drivers have to drive and P the price of a gallon of gasoline. a) Find the expression that shows the total amount of money that Driver A will save and the total amount of money that Driver B will save if both choose to get the more efficient car they are considering at the moment. b) Who is going to save more money? Does this answer depend on X and/or P? Why or why not? c) How do you explain this result? d) Explain why miles/gallon generates a nonlinear relationship if we want to know how much money is saved by increasing the efficiency of a car, but gallons/mile generates a linear relationship. For what question would gallons/mile generate a nonlinear relationship? Explain. 10. Consider the following statement: If two construction workers can build a house in 40 days, four workers can build the same house in 20 days. This statement makes two assumptions about the relationship between the number of construction workers and the time it takes to build a house. What are these assumptions? Are they good ones? Explain. 11. Consider the following eight functional relationships between X and Y: Number 1: Upward sloping and linear Number 2: Upward sloping and concave Number 3: Upward sloping and convex Number 4: Downward sloping and linear Number 5: Downward sloping and convex Number 6: Downward sloping and concave Number 7: Quadratic and concave Number 8: Quadratic and convex

4 Give one example of each relationship shown above. Your examples do not have to fit these graphs perfectly, as long as they satisfy the definitions above. 12. Consider the following four probabilistic statements. For each statement, answer how you would interpret the exact probabilities given below. a) The website fivethirtyeight, which specializes in opinion polls, forecasts that the Democratic party has an 80% probability of taking control of the House in November. b) The Weather Channels says that there is a 20% probability of precipitation for next Monday in Spartanburg. c) Home pregnancy tests are 99% accurate. d) Male condoms have a theoretical effectiveness of 98%. 13. A very sick person is rushed to the hospital. Is it more likely that this patient will die within a week or within a year? Why? 14. If I roll two dice, what is the probability that the sum of the two numbers is going to be 7? 15. Imagine that an event has a 50% probability of happening tomorrow, and a 50% probability of happening the day after tomorrow. Assume that these probabilities are independent of each other. What is the probability that this event is going to happen at least once? What is the probability that the event will happen both tomorrow and the day after tomorrow? What is the probability that this event will not happen in the next two days?

5 16. Imagine that you are working on a project that has 10 steps, and each step has a 90% probability of being successful. The project, however, will only succeed if all ten steps are successful. What is the probability that your project will succeed? 17. A hypothetical person has a 1% probability of having nightmares each night. What is the probability that this person will have at least one nightmare in a period of 100 days? 18. Imagine that the probability that each event is going to happen is p. How many events are necessary so that the probability of at least one event happening is higher than 50%? How many events are necessary so that the probability of at least one event happening is higher than 90%? 19. Suppose that 2% of all men are color-blind, and 6% of all women are color-blind. Given this information, answer the following questions: a) If I choose a person randomly, what is the probability that this person is color-blind? b) If I choose a random color-blind person, what is the probability that this person is male? c) If I choose a random color-blind person, what is the probability that this person is female? 20. (Review) Mammograms are the most frequently used screening technology to identify breast cancer. Suppose that out of all women who are over the age of 40 and are asymptomatic, 1% of them have breast cancer. We selected a random woman from this population to take the mammogram. The mammogram is 90% effective. This means that, if this woman has breast cancer, there is a 90% probability that the mammogram will correctly conclude that she has breast cancer, and if she does not have breast cancer, there is a 90% probability that the mammogram will correctly conclude that she does not have breast cancer. This woman took the mammogram and the result was positive, suggesting that she has the disease. What is the probability that she has breast cancer? 21. Define P(A B) as the probability of A being true given that B is true. a) Give two examples where P(A B) is not very different from P(B A); b) Give two examples (not mentioned in class) where P(A B) is very different from P(B A); For the next questions, use the formula for Bayes Rule to find the answers. c) When will P(A B) > P(A)? d) Assume P(A) < 0.5. When will P(A B) > 0.5? e) Assume that a person made a claim X that you consider to be extraordinary. This means of course that P(A) 0. This person, however, showed evidence E in favor of X. Assume that P(E X) = 1 (what does that mean?). If you only change your mind if your degree of belief is greater than 0.5, then what is the condition for you to change your mind in this case? Explain.

6 22. In our original breast cancer problem, the prior probability was equal to 1%, and the accuracy of the mammogram was 90% for both women with and without cancer. Suppose it is possible to develop a technology to improve the accuracy of the mammogram from 90% to 95%, but only for one group of women. In other words, we can either make the mammogram 95% accurate for women who have breast cancer or 95% accurate for women who do not have breast cancer. Given this information, answer the following questions: a) If we develop a technology that makes the mammogram 95% accurate for women with breast cancer (the accuracy for the other group remains at 90%), what is the probability that a woman with a positive result has breast cancer? b) If we develop a technology that makes the mammogram 95% accurate for women who do not have breast cancer (the accuracy for the other group remains at 90%), what is the probability that a woman with a positive result has breast cancer? c) If you had to choose between these two technologies, which one would you choose? Justify your answer. 23. In conditions of uncertainty, there are two types of mistakes that we can make: a) To believe that something (an event, statement, hypothesis, etc.) is true when it actually is not (a false positive, or an error of commission); or b) To believe that something is not true when it actually is (a false negative, or an error of omission). a) Give two examples where, according to your opinion, a false positive is more costly (i.e. worse) than a false negative; b) Give two examples where, according to your opinion, a false negative is more costly (i.e. worse) than a false positive; 24. If the prior probability is either 0 or 1, is there any type of evidence that can cause the posterior probability to be different than either 0 or 1, respectively? Explain.