My Success Assignment ‘where you want to be in ten years’ Objective Make a plan and try to see all the details. Does some research, ask questions, and consider what it’s going to take to get where you want to be.

My Success Assignment ‘where you want to be in ten years’ Objective Make a plan and try to see all the details. Does some research, ask questions, and consider what it’s going to take to get where you want to be.

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WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

WEEKLY ASSIGNMENT #2 YOU 1. Verify for the Cobb-Douglas production function P(L;K) = 1:01L:75K:25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by to double production? 1 2. Let F(x; y) = 1+ p 4 ? y2. Evaluate F(3; 1). Find and sketch the domain of F. Find the range of F. 2 3. Draw a contour map of the function showing several level curves. (a) g(x; y) = x2 ? y2 (b) s(x; y) = y=(x2 + y2) 3 4. Find the limit if it exists or show that the limit does not exist. You do not have to use the epsilon delta method so it will either be “obviously” continuous or you will have to show that it is not by finding two paths which give different results. (a) lim (x;y)!(2;?1) x2y + xy2 x2 ? y2 (b) lim (x;y)!(0;0) x4 ? 4y2 x2 + 2y2 (c) lim (x;y)!(0;0) xy p x2 + y2 4 5. The temperature T at a location in the Norther Hemisphere depends on the longitude x, the latitude y, and the time t. What are the meaning of the partial derivatives @T=@t; @T=@x; @T=@y? Moscow lies at 46:73N; 117W. Suppose that at 9 am on January 1st the wind is blowing hot air to the northeast so the air to the west and south is warm, and the air to the north and east is cooler. Would you expect fx(117; 4673; 9); fy(117; 4673; 9); ft(117; 4673; 9) to be positive negative or positive? 5 6. Find the first partial derivatives of the following functions. (a) f(x; y) = x4 + 5xy3 (b) g(x; y) = t2e?t (c) h(s; t) = ln(s + t2) (d) i(x; y) = x y (e) R(p; q) = arctan pq2 6 7. Find @z=@x and @z=@y for the following, assuming that f and g are differentiable single variable functions Hint: Your answer should use f0 and/or g0. z = f(x)g(y) ; z = f(x=y) 7

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Take Home Exam 3: Special Note Before Starting the Exam: If you scan your solutions to the exam and save it as a pdf or image file and put it on dropbox and I can not read it or open it, you will not receive credit for the exam. Furthermore, if you write the solutions up in word, latex ect. and give me a print out, which does not include all the pages you will not get credit for the missing pages. Also if your folder on dropbox is not clearly labeled and I can not find your exam then you will not get credit for the exam. Finally, please make sure you put your name on the exam!! Math 2100 Exam 3, Out of Class, Due by December 8th, 2015 at 5:00 pm. Name: Problem 1. (15 points) A random variable is said to have the (standard) Cauchy distribution if its PDF is given by f (x) = 1 π 1 1+ x2 , −∞< x <∞ This problem uses computer simulations to demonstrate that a) samples from this distribution often have extreme outliers (a consequence of the heavy tails of the distribution), and b) the sample mean is prone to the same type of outliers. Below is a graph of the pdf a) (5 points) The R commands x=rcauchy(500); summary(x) generate a random sample of size 500 from the Cauchy distribution and display the sample’s five number summary; Report the five number summary and the interquartile range, and comment on whether or not the smallest and largest numbers generated from this sample of 500 are outliers. Repeat this 10 times. b) (5 points) The R commands m=matrix(rcauchy(50000), nrow=500); xb=apply(m,1,mean);summary(xb) generate the matrix m that has 500 rows, each of which is a sample of size n=100 from the Cauchy distribution, compute the 500 sample means and store them in xb. and display the five number summary xb. Repeat these commands 10 times, and report the 10 sets of five number summaries. Compare with the 10 sets of five number summaries from part (a), and comment on whether or not the distribution of the averages seems to be more prone to extreme outliers as that of the individual observations. c) (5 points) Why does this happen? (hint: try to calculate E(X) and V(X) for this distribution) and does the LLN and CLT apply for samples from a Cauchy distribution? Hint: E(X) is undefined for this distribution unless you use the Cauchy Principle Value as such for the mean lim a→∞ xf (x)dx −a a∫ In addition x2 1+ x2 dx = x2 +1−1 1+ x2 dx = 1− 1 1+ x2 " # $ % & ' ∫ ∫ ∫ dx 1 1+ x2 dx = tan−1 ∫ x +C Problem 2. (5 points) A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna. If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert's claim? Use a 0.05 level of significance. Problem 3. (10 points) A coin is tossed 20 times, resulting in 5 heads. Is this sufficient evidence to reject the hypothesis that the coin is balanced in favor of the alternative that heads occur less than 50% of the time (essentially is this significant evidence to claim that the coin is unbalanced in favor of tails)? Use a 0.05 level of significance. Problem 4. (25 points) Since the chemical benzene may cause cancer, the federal government has set the maximum allowable benzene concentration in the workplace at 1 part per million (1 ppm) Suppose that a steel manufacturing plant is under investigation for possible violations regarding benzene level. The Occupational Safety and Health Administration (OSHA) will analyze 14 air samples over a one-month period. Assume normality of the population from which the samples were drawn. a) (3 points) What is an appropriate null hypothesis for this scenario? (Give this in symbols) b) (3 points) What is an appropriate alternative hypothesis for this scenario? (Give this in symbols) c) (3 points) What kind of hypothesis test is this: left-tailed, right-tailed or two-tailed? Explain how you picked your answer. d) (3 points) Is this a one-sample t-test or a one-sample test using a normal distribution? Explain how you picked your answer. e) (4 points) If the test using this sample of size 14 is to be done at the 1% significance level, calculate the critical value(s) and describe the rejection region(s) for the test statistic. Show your work. f) (5 points) OHSA finds the following for their sample of size 14: a mean benzene level of 1.51 ppm and a standard deviation of 1.415 ppm. What should be concluded at the 1% significance level? Support your answer with calculation(s) and reasoning. g) (4 points) Calculate the p-value for this test and verify that this answer would lead to the same conclusion you made in part f. Problem 5. (15 points) A normally distributed random variable Y possesses a mean of μ = 20 and a standard deviation of σ = 5. A random sample of n = 31 observations is to be selected. Let X be the sample average. (X in this problem is really x _ ) a)(5 points) Describe the sampling distribution of X (i.e. describe the distribution of X and give μx, σx ) b) (5 points) Find the z-score of x = 22 c) (5 points) Find P(X ≥ 22) = Problem 6. (10 points) A restaurants receipts show that the cost of customers' dinners has a distribution with a mean of $54 and a standard deviation of $18. What is the probability that the next 100 customers will spend a total of at least $5800 on dinner? Problem 7. (10 points) The operations manager of a large production plant would like to estimate the mean amount of time a worker takes to assemble a new electronic component. Assume that the standard deviation of this assembly time is 3.6 minutes and is normally distributed. a) (3 points) After observing 120 workers assembling similar devices, the manager noticed that their average time was 16.2 minutes. Construct a 92% confidence interval for the mean assembly time. b) (2 points) How many workers should be involved in this study in order to have the mean assembly time estimated up to ± 15 seconds with 92% confidence? c) (5 points) Construct a 92% confidence interval if instead of observing 120 workers assembling similar devices, rather the manager observes 25 workers and notice their average time was 16.2 minutes with a standard deviation of 4.0 minutes. Problem 8. (10 points): A manufacturer of candy must monitor the temperature at which the candies are baked. Too much variation will cause inconsistency in the taste of the candy. Past records show that the standard deviation of the temperature has been 1.2oF . A random sample of 30 batches of candy is selected, and the sample standard deviation of the temperature is 2.1oF . a. (5 points) At the 0.05 level of significance, is there evidence that the population standard deviation has increased above 1.2oF ? b. (3 points) What assumption do you need to make in order to perform this test? c. (2 points) Compute the p-value in (a) and interpret its meaning.

Take Home Exam 3: Special Note Before Starting the Exam: If you scan your solutions to the exam and save it as a pdf or image file and put it on dropbox and I can not read it or open it, you will not receive credit for the exam. Furthermore, if you write the solutions up in word, latex ect. and give me a print out, which does not include all the pages you will not get credit for the missing pages. Also if your folder on dropbox is not clearly labeled and I can not find your exam then you will not get credit for the exam. Finally, please make sure you put your name on the exam!! Math 2100 Exam 3, Out of Class, Due by December 8th, 2015 at 5:00 pm. Name: Problem 1. (15 points) A random variable is said to have the (standard) Cauchy distribution if its PDF is given by f (x) = 1 π 1 1+ x2 , −∞< x <∞ This problem uses computer simulations to demonstrate that a) samples from this distribution often have extreme outliers (a consequence of the heavy tails of the distribution), and b) the sample mean is prone to the same type of outliers. Below is a graph of the pdf a) (5 points) The R commands x=rcauchy(500); summary(x) generate a random sample of size 500 from the Cauchy distribution and display the sample’s five number summary; Report the five number summary and the interquartile range, and comment on whether or not the smallest and largest numbers generated from this sample of 500 are outliers. Repeat this 10 times. b) (5 points) The R commands m=matrix(rcauchy(50000), nrow=500); xb=apply(m,1,mean);summary(xb) generate the matrix m that has 500 rows, each of which is a sample of size n=100 from the Cauchy distribution, compute the 500 sample means and store them in xb. and display the five number summary xb. Repeat these commands 10 times, and report the 10 sets of five number summaries. Compare with the 10 sets of five number summaries from part (a), and comment on whether or not the distribution of the averages seems to be more prone to extreme outliers as that of the individual observations. c) (5 points) Why does this happen? (hint: try to calculate E(X) and V(X) for this distribution) and does the LLN and CLT apply for samples from a Cauchy distribution? Hint: E(X) is undefined for this distribution unless you use the Cauchy Principle Value as such for the mean lim a→∞ xf (x)dx −a a∫ In addition x2 1+ x2 dx = x2 +1−1 1+ x2 dx = 1− 1 1+ x2 " # $ % & ' ∫ ∫ ∫ dx 1 1+ x2 dx = tan−1 ∫ x +C Problem 2. (5 points) A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna. If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert's claim? Use a 0.05 level of significance. Problem 3. (10 points) A coin is tossed 20 times, resulting in 5 heads. Is this sufficient evidence to reject the hypothesis that the coin is balanced in favor of the alternative that heads occur less than 50% of the time (essentially is this significant evidence to claim that the coin is unbalanced in favor of tails)? Use a 0.05 level of significance. Problem 4. (25 points) Since the chemical benzene may cause cancer, the federal government has set the maximum allowable benzene concentration in the workplace at 1 part per million (1 ppm) Suppose that a steel manufacturing plant is under investigation for possible violations regarding benzene level. The Occupational Safety and Health Administration (OSHA) will analyze 14 air samples over a one-month period. Assume normality of the population from which the samples were drawn. a) (3 points) What is an appropriate null hypothesis for this scenario? (Give this in symbols) b) (3 points) What is an appropriate alternative hypothesis for this scenario? (Give this in symbols) c) (3 points) What kind of hypothesis test is this: left-tailed, right-tailed or two-tailed? Explain how you picked your answer. d) (3 points) Is this a one-sample t-test or a one-sample test using a normal distribution? Explain how you picked your answer. e) (4 points) If the test using this sample of size 14 is to be done at the 1% significance level, calculate the critical value(s) and describe the rejection region(s) for the test statistic. Show your work. f) (5 points) OHSA finds the following for their sample of size 14: a mean benzene level of 1.51 ppm and a standard deviation of 1.415 ppm. What should be concluded at the 1% significance level? Support your answer with calculation(s) and reasoning. g) (4 points) Calculate the p-value for this test and verify that this answer would lead to the same conclusion you made in part f. Problem 5. (15 points) A normally distributed random variable Y possesses a mean of μ = 20 and a standard deviation of σ = 5. A random sample of n = 31 observations is to be selected. Let X be the sample average. (X in this problem is really x _ ) a)(5 points) Describe the sampling distribution of X (i.e. describe the distribution of X and give μx, σx ) b) (5 points) Find the z-score of x = 22 c) (5 points) Find P(X ≥ 22) = Problem 6. (10 points) A restaurants receipts show that the cost of customers' dinners has a distribution with a mean of $54 and a standard deviation of $18. What is the probability that the next 100 customers will spend a total of at least $5800 on dinner? Problem 7. (10 points) The operations manager of a large production plant would like to estimate the mean amount of time a worker takes to assemble a new electronic component. Assume that the standard deviation of this assembly time is 3.6 minutes and is normally distributed. a) (3 points) After observing 120 workers assembling similar devices, the manager noticed that their average time was 16.2 minutes. Construct a 92% confidence interval for the mean assembly time. b) (2 points) How many workers should be involved in this study in order to have the mean assembly time estimated up to ± 15 seconds with 92% confidence? c) (5 points) Construct a 92% confidence interval if instead of observing 120 workers assembling similar devices, rather the manager observes 25 workers and notice their average time was 16.2 minutes with a standard deviation of 4.0 minutes. Problem 8. (10 points): A manufacturer of candy must monitor the temperature at which the candies are baked. Too much variation will cause inconsistency in the taste of the candy. Past records show that the standard deviation of the temperature has been 1.2oF . A random sample of 30 batches of candy is selected, and the sample standard deviation of the temperature is 2.1oF . a. (5 points) At the 0.05 level of significance, is there evidence that the population standard deviation has increased above 1.2oF ? b. (3 points) What assumption do you need to make in order to perform this test? c. (2 points) Compute the p-value in (a) and interpret its meaning.

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. Read the article on Lean Production at Jaguar (when it used to be part of Ford) which is provided at the link: http://businesscasestudies.co.uk/jaguar/lean-production/introduction.html#axzz2SrprdGnx Once you read this case study answer the following questions: (a) A generic pharmaceutical company wanted to implement Lean Manufacturing in their manufacturing process. They hired a project manager X from the automotive industry who had extensive experience implementing lean. Extract some lessons learned and best practices from the Jaguar case study that pharmaceutical company could implement at their plant once the new project manager in charge of lean came onboard. Explain each point in detail. Also, state any additional steps that project manager X could take to implement lean at pharmaceutical company. [10 points].

. Read the article on Lean Production at Jaguar (when it used to be part of Ford) which is provided at the link: http://businesscasestudies.co.uk/jaguar/lean-production/introduction.html#axzz2SrprdGnx Once you read this case study answer the following questions: (a) A generic pharmaceutical company wanted to implement Lean Manufacturing in their manufacturing process. They hired a project manager X from the automotive industry who had extensive experience implementing lean. Extract some lessons learned and best practices from the Jaguar case study that pharmaceutical company could implement at their plant once the new project manager in charge of lean came onboard. Explain each point in detail. Also, state any additional steps that project manager X could take to implement lean at pharmaceutical company. [10 points].

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http://www.econlib.org/library/Smith/smWN1.html#B.I,%20Ch.1,%20Of%20the%20Division%20of%20Labor What does Smith mean by division of labor, and how does it affect production? A. He means that each person does their own work to benefit themselves by creating goods. This creates well-crafted goods. B. He argues that in order to become more efficient, we need to put everyone in the same workhouses and eliminate division. C. He says that the division of labor provides farmers with the opportunity to become involved in manufacturing. D. He means that each person makes one small part of a good very quickly, but this is bad for the quality of production overall. E. He means that by having each individual specialize in one thing, they can work together to create products more efficiently and effectively. Which of the following is NOT an example of the circumstances by which the division of labor improves efficiency? A. A doll-making company stops allowing each employee to make one whole doll each and instead appoints each employee to create one part of the doll. B. A family of rug makers buys a loom to speed up their production. C. A mechanic opens a new shop to be nearer to the market. D. A factory changes the responsibilities of its employees so that one group handles heavy boxes and the other group does precision sewing. E. A baker who used to make a dozen cookies at a time buys a giant mixer and oven that enable him to make 20 dozen cookies at a time. Considering the global system of states, what do you think the allegory of the pins has to offer? A. It suggests that there could be a natural harmony of interests among states because they can divide labor among themselves to the benefit of everyone. B. It suggests that states can never be secure enough to cooperate because every state is equally capable of producing the same things. C. It suggests that a central authority is necessary to help states cooperate, in the same way that a manager oversees operations at a factory. D. The allegory of the pins is a great way to think about how wars come about, because states won’t cooperate with each other like pin-makers do. E. The allegory of the pins shows us that there is no natural harmony of interests between states. Smith sees the development of industry, technology, and the division of labor as A. generally positive but not progressive. The lives of many people may improve, but the world will generally stay the same. B. generally positive and progressive. The world is improving because of these changes, and it will continue to improve. C. generally negative. The creation of new technologies and the division of labor are harmful to all humans, both the wealthy and the poor. D. generally negative. The creation of the division of labor only benefits the wealthy at the expense of the poor. E. both positive and negative. Smith thinks that technology hurts us, while the division of labor helps society progress and develop. http://www.youtube.com/watch?v=RUwS1uAdUcI What point is Hans Rosling trying to make when he describes the global health pre-test? A. He is trying to show how the average person has no idea of the true state of global health. B. He is trying to illustrate how we tend to carry around outdated notions about the state of global health. C. He is trying to make us see that the less-developed countries are far worse off than we ever thought. D. He is trying to drive home the idea that global health has not improved over time despite foreign aid and improvements in medicine. E. He is trying to warn us about the rapid growth in world population. Rosling shows us that we tend to think about global health in terms of “we and them.” Who are the “we” and who are the “them”? A. “We” refers to academics, students, and scholars; “them” refers to the uneducated. B. “We” refers to the average person; “them” refers to politicians and global leaders. C. “We” refers to the wealthy; “them” refers to the poor. D. “We” refers to the Western world; “them” refers to the Third World. E. “We” refers to students; “them” refers to professors. In the life expectancy and fertility rate demonstration, what do the statistics reveal? A. Over time, developed countries produced small families and long lives, whereas developing countries produced large families and short lives. B. The world today looks much like it did in 1962 despite our attempts to help poorer countries develop. C. All countries in the world, even the poorer ones, are trending toward longer lives and smaller families. D. Developed countries are trending toward smaller families but shorter lives. E. All countries tend to make gains and losses in fertility and lifespan, but in the long run there is no significant change. What point does Rosling make about life expectancy in Vietnam as compared to the United States? To what does he attribute the change? A. He indicates that economic change preceded social change. B. He suggests that markets and free trade resulted in the increase in life expectancy. C. He says that the data indicates that the Vietnam War contributed to the decrease in life expectancy during that time, but that it recovered shortly thereafter. D. He says that social change in Asia preceded economic change, and life expectancy in Vietnam increased despite the war. E. He indicates that Vietnam was equal to the United States in life expectancy before the war. According to Rosling, how are regional statistics about child survival rates and GDP potentially misleading? A. Countries have an incentive to lie about the actual survival rates because they want foreign assistance. B. Statistics for the individual countries in a region are often vastly different. C. Regional statistics give us a strong sense of how we can understand development within one region, but it does not allow us to compare across regions. D. The data available over time and from countries within regions is often poorly collected and incomplete. E. Child survival rates cannot be compared regionally, since each culture has a different sense of how important children are. What is Rosling’s main point about statistical databases? A. The data is available but not readily accessible, so we need to create networks to solve that problem. B. The data that comes from these databases is often flawed and unreliable. C. It doesn’t matter whether we have access to these databases because the data can’t be used in an interesting way. D. Statistics can’t tell us very much, but we should do our best to make use of the information we do have. E. The information that could be true is too hard to sort out from what isn’t true because we don’t know how strong the data really is. http://www.marxists.org/archive/lenin/works/1916/imp-hsc/ch10.htm#v22zz99h-298-GUESS Click the link at left to read Chapter 10 of Imperialism, The Highest Stage of Capitalism, then answer the questions below. According to Lenin, what is the fundamental source of a monopoly? A. It is a natural effect of human behavior. B. It is the result of governments and police systems. C. Its source is rooted in democracy. D. It comes from the concentration of production at a high stage. E. It is what follows a socialist system. What are the principal types or manifestations of monopoly capitalism? A. Monopolistic capitalist associations like cartels, syndicates and trusts; and monopolies as a result of colonial policy. B. Monopolization of raw materials and monopolization of finance capital. C. Monopolization of governing structures and monopolies of oligarchies. D. Monopolist capitalist associations like cartels, syndicates and trusts; and monopolies as a result of colonial policy AND monopolization of raw materials and monopolization of finance capital. E. Monopolization of raw materials and monopolization of finance capital AND monopolization of governing structures and monopolies of oligarchies. What is the definition of a rentier state according to Lenin? A. A state that colonizes other states. B. A state whose bourgeoisie live off the export of capital. C. A poor state. D. A wealthy state. E. A colonized state. Overall Lenin’s analysis of the state of capitalism is concerned with: A. The interactions between states. B. The interactions within states. C. The ownership of industry and organizations. D. The interactions within states AND the ownership of industry and organizations. E. All of these options. http://view.vzaar.com/1194665/flashplayer Watch the video at left, and then answer the questions below. The Marshall Plan was developed by the United States after World War II. What was its purpose? A. to feed the hungry of Europe B. to stem the spread of communism C. to maintain an American military presence in Europe D. to feed the hungry of Europe AND to stem the spread of communism E. to stem the spread of communism AND to maintain an American military presence in Europe What kind of aid was sent at first? A. foods, fertilizers, and machines for agriculture B. books, paper, and radios for education C. clothing, medical supplies, and construction equipment D. mostly cash in the form of loans and grants E. people with business expertise to help develop the economy What kind of aid did the United States send to Greece to help its farmers? A. tractors B. mules C. seeds D. fertilizer E. all of these options What was one way that the United States influenced public opinion in Italy during the elections described in the video? A. The United States provided significant food aid to Italy so that the Italians would be inclined to vote against the Communists. B. The Italians had been impressed by the strength and loyalty of the American soldiers, and were inclined to listen to them during the elections. C. There was a large number of young Italians who followed American fashion and culture. D. Italian immigrants in the United States wrote letters to their families in Italy urging them not to vote for Communists. E. The Greeks showed the Italians how much the Americans had helped them, warning that supporting a Communist candidate would mean sacrificing American aid. How did Pope Pius XII undermine the strength of the Communist Party in Italy? A. He encouraged Italians to go out and vote. B. He warned that the Communist Party would legalize abortion. C. He excommunicated many members of the Communist Party. D. He made a speech in support of capitalism. E. He declared that Communists should not be baptized. http://www.youtube.com/watch?v=KVhWqwnZ1eM Use the video at left to answer the questions below. Hans Rosling shares how his students discuss “we” versus “them.” To whom are his students referring? A. the United States and Mexico B. Christians and Muslims C. Democrats and Republicans D. Europe and Asia E. none of these options According to Rosling, what factors contribute to a better quality of life for people in developing countries? A. family planning B. soap and water C. investment D. vaccinations E. all of these options Using his data, Rosling demonstrates a great shift in Mexico. What change does his data demonstrate? A. a decrease in drug usage B. a decrease in the number of jobs available C. an increase in average life expenctancy D. an increase in the rate of violent crime E. all of these options Instead of “developing” and “developed,” Rosling divides countries into four categories. Which of the following is NOT one of them? A. high-income countries B. middle-income countries C. low-income countries D. no-income countries E. collapsing countries Rosling discusses the increased life expectancy in both China and the United States. How are the situations different? A. The U.S. and China are on different continents. B. The life expectancy in China rose much higher than it did in the U.S. C. China first expanded its life expectancy and then grew economically, whereas the U.S. did the reverse. D. Average income and life expectancy steadily increased in the U.S., but they steadily decreased in China. E. all of these options Rosling shows a chart that demonstrates the regional income distribution of the world from 1970 to 2015. During that time, what has happened in South and East Asia? A. Money has flowed out of Asia to developing countries in Africa. B. The average income of citizens of South and East Asia has increased over the last 30 years. C. The average income of citizens of South and East Asia has decreased over the last 30 years. D. The average income of citizens of South and East Asia has surpassed that of Europe and North America. E. There has been no change. Click here to access GapMinder, the data visualizer that Hans Rosling uses. In 2010, which of the following countries had both a higher per-capita GDP and a higher life expectancy than the United States? A. France B. Japan C. Denmark D. Singapore E. Kuwait http://www.garretthardinsociety.org/articles/art_tragedy_of_the_commons.html http://www.youtube.com/watch?v=8a4S23uXIcM The Tragedy of the Commons What is the rough definition of the “commons” given in the article? A. any private property on which others trespass B. behavior that everyone considers to be normal C. a cow that lives in a herd D. government-administered benefits, like unemployment or Social Security E. a shared resource What does Hardin mean by describing pollution as a reverse tragedy of the commons? A. Rather than causing a problem, it resolves a problem. B. Pollution costs us money rather than making us money. C. We are putting something into the commons rather than removing something from it. D. It starts at the other end of the biological pyramid. E. Humans see less of it as time goes on. Hardin says “the air and waters surrounding us cannot readily be fenced, and so the tragedy of the commons as a cesspool must be prevented by different means.” What are those means? A. establishing more international treaties to protect the environment B. using laws or taxes to make the polluter pay for pollution C. punishing consumers for generating waste D. raising awareness about environmental issues E. developing greener products Pacific Garbage Dump According to the news report, what percent of the Gyre is made of plastic? A. 50 percent B. 60 percent C. 70 percent D. 80 percent E. 90 percent Where does the majority of the plastic in the Gyre come from? A. barges that dump trash in the ocean B. storm drains from land C. people throwing litter off boats into the ocean D. remnants from movie sets filmed at sea E. fishing boats processing their catch What does Charles Moore mean by the “throwaway concept”? A. the habitual use of disposable plastic packaging B. the mistaken view that marine ecosystems are infinitely renewable C. a general lack of interest in recycling D. the willingness to discard effective but small-scale environmental policies in deference to broader E. people throwing away their lives in pursuit of money In what way does the Great Pacific Gyre represent issues like global warming a tragedy of the commons? A. because all the plastic trash in it comes from the United States B. because it kills the albatross and makes it impossible for them to reproduce C. surbecause the countries rounding the Pacific Ocean are polluting the water in a way that affects the quality of the resource for all, but no one is specifically accountable for it D. because it causes marine life to compete for increasingly scarce nutrients in the ocean E. because nations in the region all collectively agreed to dump their trash in the Pacific http://www.npr.org/news/specials/climate/video/ http://ngm.nationalgeographic.com/climateconnections/climate-map http://www.npr.org/news/specials/climate/video/wildchronicles.html Use the links provided at left to answer the questions below. Global Warming: It’s All About Carbon How does carbon give us fuel? A. When you burn things that contain carbon the bonds break, giving off energy. B. Burning things creates carbon out of other elements as a result of combustion. C. Carbon is created after oxygen and hydrogen get released. D. Carbon bonds are created thereby giving off energy. E. Carbon is made into fuel by refining oil. National Geographic Climate Map What geographic areas have seen the most significant changes in temperature? A. The African continent. B. The Pacific Ocean. C. The Atlantic Ocean. D. The Arctic Ocean. E. The Indian Ocean. Why does it matter that rain fall steadily rather than in downpours? A. For those countries accustomed to steady rain fall, downpours are actually more efficient ways to catch water. B. Downpours in regions accustomed to steady fall makes them more prone to flooding and damage. C. In general, as long as regions get either steady fall or downpours most things will stay the same. D. Downpours are always more beneficial to crop growth than steady rain. E. Steady rain is always more beneficial to crop growth than downpours. Climate Change Threatens Kona Coffee What is unique about the climate in Hawaii, making it a good place to grow coffee? A. The elevation is high, the nights are cool and the days are humid. B. The elevation is low, the nights are warm and the days are dry. C. The elevation is high, the nights are warm and the days are dry. D. The elevation is low, the nights are cool and the days are dry. E. The elevation is high, the nights are warm and the days are humid. What specific temperature pattern have experts noted about the region where Kona coffee is grown in Hawaii? A. There has been no significant change but the bean production has dropped. B. The nights have warmed up, even though the days have cooled. C. There has been an increase in bean production with the change in climate. D. The nights have cooled even more so than before. E. There has been universally hot days all the way around.

http://www.econlib.org/library/Smith/smWN1.html#B.I,%20Ch.1,%20Of%20the%20Division%20of%20Labor What does Smith mean by division of labor, and how does it affect production? A. He means that each person does their own work to benefit themselves by creating goods. This creates well-crafted goods. B. He argues that in order to become more efficient, we need to put everyone in the same workhouses and eliminate division. C. He says that the division of labor provides farmers with the opportunity to become involved in manufacturing. D. He means that each person makes one small part of a good very quickly, but this is bad for the quality of production overall. E. He means that by having each individual specialize in one thing, they can work together to create products more efficiently and effectively. Which of the following is NOT an example of the circumstances by which the division of labor improves efficiency? A. A doll-making company stops allowing each employee to make one whole doll each and instead appoints each employee to create one part of the doll. B. A family of rug makers buys a loom to speed up their production. C. A mechanic opens a new shop to be nearer to the market. D. A factory changes the responsibilities of its employees so that one group handles heavy boxes and the other group does precision sewing. E. A baker who used to make a dozen cookies at a time buys a giant mixer and oven that enable him to make 20 dozen cookies at a time. Considering the global system of states, what do you think the allegory of the pins has to offer? A. It suggests that there could be a natural harmony of interests among states because they can divide labor among themselves to the benefit of everyone. B. It suggests that states can never be secure enough to cooperate because every state is equally capable of producing the same things. C. It suggests that a central authority is necessary to help states cooperate, in the same way that a manager oversees operations at a factory. D. The allegory of the pins is a great way to think about how wars come about, because states won’t cooperate with each other like pin-makers do. E. The allegory of the pins shows us that there is no natural harmony of interests between states. Smith sees the development of industry, technology, and the division of labor as A. generally positive but not progressive. The lives of many people may improve, but the world will generally stay the same. B. generally positive and progressive. The world is improving because of these changes, and it will continue to improve. C. generally negative. The creation of new technologies and the division of labor are harmful to all humans, both the wealthy and the poor. D. generally negative. The creation of the division of labor only benefits the wealthy at the expense of the poor. E. both positive and negative. Smith thinks that technology hurts us, while the division of labor helps society progress and develop. http://www.youtube.com/watch?v=RUwS1uAdUcI What point is Hans Rosling trying to make when he describes the global health pre-test? A. He is trying to show how the average person has no idea of the true state of global health. B. He is trying to illustrate how we tend to carry around outdated notions about the state of global health. C. He is trying to make us see that the less-developed countries are far worse off than we ever thought. D. He is trying to drive home the idea that global health has not improved over time despite foreign aid and improvements in medicine. E. He is trying to warn us about the rapid growth in world population. Rosling shows us that we tend to think about global health in terms of “we and them.” Who are the “we” and who are the “them”? A. “We” refers to academics, students, and scholars; “them” refers to the uneducated. B. “We” refers to the average person; “them” refers to politicians and global leaders. C. “We” refers to the wealthy; “them” refers to the poor. D. “We” refers to the Western world; “them” refers to the Third World. E. “We” refers to students; “them” refers to professors. In the life expectancy and fertility rate demonstration, what do the statistics reveal? A. Over time, developed countries produced small families and long lives, whereas developing countries produced large families and short lives. B. The world today looks much like it did in 1962 despite our attempts to help poorer countries develop. C. All countries in the world, even the poorer ones, are trending toward longer lives and smaller families. D. Developed countries are trending toward smaller families but shorter lives. E. All countries tend to make gains and losses in fertility and lifespan, but in the long run there is no significant change. What point does Rosling make about life expectancy in Vietnam as compared to the United States? To what does he attribute the change? A. He indicates that economic change preceded social change. B. He suggests that markets and free trade resulted in the increase in life expectancy. C. He says that the data indicates that the Vietnam War contributed to the decrease in life expectancy during that time, but that it recovered shortly thereafter. D. He says that social change in Asia preceded economic change, and life expectancy in Vietnam increased despite the war. E. He indicates that Vietnam was equal to the United States in life expectancy before the war. According to Rosling, how are regional statistics about child survival rates and GDP potentially misleading? A. Countries have an incentive to lie about the actual survival rates because they want foreign assistance. B. Statistics for the individual countries in a region are often vastly different. C. Regional statistics give us a strong sense of how we can understand development within one region, but it does not allow us to compare across regions. D. The data available over time and from countries within regions is often poorly collected and incomplete. E. Child survival rates cannot be compared regionally, since each culture has a different sense of how important children are. What is Rosling’s main point about statistical databases? A. The data is available but not readily accessible, so we need to create networks to solve that problem. B. The data that comes from these databases is often flawed and unreliable. C. It doesn’t matter whether we have access to these databases because the data can’t be used in an interesting way. D. Statistics can’t tell us very much, but we should do our best to make use of the information we do have. E. The information that could be true is too hard to sort out from what isn’t true because we don’t know how strong the data really is. http://www.marxists.org/archive/lenin/works/1916/imp-hsc/ch10.htm#v22zz99h-298-GUESS Click the link at left to read Chapter 10 of Imperialism, The Highest Stage of Capitalism, then answer the questions below. According to Lenin, what is the fundamental source of a monopoly? A. It is a natural effect of human behavior. B. It is the result of governments and police systems. C. Its source is rooted in democracy. D. It comes from the concentration of production at a high stage. E. It is what follows a socialist system. What are the principal types or manifestations of monopoly capitalism? A. Monopolistic capitalist associations like cartels, syndicates and trusts; and monopolies as a result of colonial policy. B. Monopolization of raw materials and monopolization of finance capital. C. Monopolization of governing structures and monopolies of oligarchies. D. Monopolist capitalist associations like cartels, syndicates and trusts; and monopolies as a result of colonial policy AND monopolization of raw materials and monopolization of finance capital. E. Monopolization of raw materials and monopolization of finance capital AND monopolization of governing structures and monopolies of oligarchies. What is the definition of a rentier state according to Lenin? A. A state that colonizes other states. B. A state whose bourgeoisie live off the export of capital. C. A poor state. D. A wealthy state. E. A colonized state. Overall Lenin’s analysis of the state of capitalism is concerned with: A. The interactions between states. B. The interactions within states. C. The ownership of industry and organizations. D. The interactions within states AND the ownership of industry and organizations. E. All of these options. http://view.vzaar.com/1194665/flashplayer Watch the video at left, and then answer the questions below. The Marshall Plan was developed by the United States after World War II. What was its purpose? A. to feed the hungry of Europe B. to stem the spread of communism C. to maintain an American military presence in Europe D. to feed the hungry of Europe AND to stem the spread of communism E. to stem the spread of communism AND to maintain an American military presence in Europe What kind of aid was sent at first? A. foods, fertilizers, and machines for agriculture B. books, paper, and radios for education C. clothing, medical supplies, and construction equipment D. mostly cash in the form of loans and grants E. people with business expertise to help develop the economy What kind of aid did the United States send to Greece to help its farmers? A. tractors B. mules C. seeds D. fertilizer E. all of these options What was one way that the United States influenced public opinion in Italy during the elections described in the video? A. The United States provided significant food aid to Italy so that the Italians would be inclined to vote against the Communists. B. The Italians had been impressed by the strength and loyalty of the American soldiers, and were inclined to listen to them during the elections. C. There was a large number of young Italians who followed American fashion and culture. D. Italian immigrants in the United States wrote letters to their families in Italy urging them not to vote for Communists. E. The Greeks showed the Italians how much the Americans had helped them, warning that supporting a Communist candidate would mean sacrificing American aid. How did Pope Pius XII undermine the strength of the Communist Party in Italy? A. He encouraged Italians to go out and vote. B. He warned that the Communist Party would legalize abortion. C. He excommunicated many members of the Communist Party. D. He made a speech in support of capitalism. E. He declared that Communists should not be baptized. http://www.youtube.com/watch?v=KVhWqwnZ1eM Use the video at left to answer the questions below. Hans Rosling shares how his students discuss “we” versus “them.” To whom are his students referring? A. the United States and Mexico B. Christians and Muslims C. Democrats and Republicans D. Europe and Asia E. none of these options According to Rosling, what factors contribute to a better quality of life for people in developing countries? A. family planning B. soap and water C. investment D. vaccinations E. all of these options Using his data, Rosling demonstrates a great shift in Mexico. What change does his data demonstrate? A. a decrease in drug usage B. a decrease in the number of jobs available C. an increase in average life expenctancy D. an increase in the rate of violent crime E. all of these options Instead of “developing” and “developed,” Rosling divides countries into four categories. Which of the following is NOT one of them? A. high-income countries B. middle-income countries C. low-income countries D. no-income countries E. collapsing countries Rosling discusses the increased life expectancy in both China and the United States. How are the situations different? A. The U.S. and China are on different continents. B. The life expectancy in China rose much higher than it did in the U.S. C. China first expanded its life expectancy and then grew economically, whereas the U.S. did the reverse. D. Average income and life expectancy steadily increased in the U.S., but they steadily decreased in China. E. all of these options Rosling shows a chart that demonstrates the regional income distribution of the world from 1970 to 2015. During that time, what has happened in South and East Asia? A. Money has flowed out of Asia to developing countries in Africa. B. The average income of citizens of South and East Asia has increased over the last 30 years. C. The average income of citizens of South and East Asia has decreased over the last 30 years. D. The average income of citizens of South and East Asia has surpassed that of Europe and North America. E. There has been no change. Click here to access GapMinder, the data visualizer that Hans Rosling uses. In 2010, which of the following countries had both a higher per-capita GDP and a higher life expectancy than the United States? A. France B. Japan C. Denmark D. Singapore E. Kuwait http://www.garretthardinsociety.org/articles/art_tragedy_of_the_commons.html http://www.youtube.com/watch?v=8a4S23uXIcM The Tragedy of the Commons What is the rough definition of the “commons” given in the article? A. any private property on which others trespass B. behavior that everyone considers to be normal C. a cow that lives in a herd D. government-administered benefits, like unemployment or Social Security E. a shared resource What does Hardin mean by describing pollution as a reverse tragedy of the commons? A. Rather than causing a problem, it resolves a problem. B. Pollution costs us money rather than making us money. C. We are putting something into the commons rather than removing something from it. D. It starts at the other end of the biological pyramid. E. Humans see less of it as time goes on. Hardin says “the air and waters surrounding us cannot readily be fenced, and so the tragedy of the commons as a cesspool must be prevented by different means.” What are those means? A. establishing more international treaties to protect the environment B. using laws or taxes to make the polluter pay for pollution C. punishing consumers for generating waste D. raising awareness about environmental issues E. developing greener products Pacific Garbage Dump According to the news report, what percent of the Gyre is made of plastic? A. 50 percent B. 60 percent C. 70 percent D. 80 percent E. 90 percent Where does the majority of the plastic in the Gyre come from? A. barges that dump trash in the ocean B. storm drains from land C. people throwing litter off boats into the ocean D. remnants from movie sets filmed at sea E. fishing boats processing their catch What does Charles Moore mean by the “throwaway concept”? A. the habitual use of disposable plastic packaging B. the mistaken view that marine ecosystems are infinitely renewable C. a general lack of interest in recycling D. the willingness to discard effective but small-scale environmental policies in deference to broader E. people throwing away their lives in pursuit of money In what way does the Great Pacific Gyre represent issues like global warming a tragedy of the commons? A. because all the plastic trash in it comes from the United States B. because it kills the albatross and makes it impossible for them to reproduce C. surbecause the countries rounding the Pacific Ocean are polluting the water in a way that affects the quality of the resource for all, but no one is specifically accountable for it D. because it causes marine life to compete for increasingly scarce nutrients in the ocean E. because nations in the region all collectively agreed to dump their trash in the Pacific http://www.npr.org/news/specials/climate/video/ http://ngm.nationalgeographic.com/climateconnections/climate-map http://www.npr.org/news/specials/climate/video/wildchronicles.html Use the links provided at left to answer the questions below. Global Warming: It’s All About Carbon How does carbon give us fuel? A. When you burn things that contain carbon the bonds break, giving off energy. B. Burning things creates carbon out of other elements as a result of combustion. C. Carbon is created after oxygen and hydrogen get released. D. Carbon bonds are created thereby giving off energy. E. Carbon is made into fuel by refining oil. National Geographic Climate Map What geographic areas have seen the most significant changes in temperature? A. The African continent. B. The Pacific Ocean. C. The Atlantic Ocean. D. The Arctic Ocean. E. The Indian Ocean. Why does it matter that rain fall steadily rather than in downpours? A. For those countries accustomed to steady rain fall, downpours are actually more efficient ways to catch water. B. Downpours in regions accustomed to steady fall makes them more prone to flooding and damage. C. In general, as long as regions get either steady fall or downpours most things will stay the same. D. Downpours are always more beneficial to crop growth than steady rain. E. Steady rain is always more beneficial to crop growth than downpours. Climate Change Threatens Kona Coffee What is unique about the climate in Hawaii, making it a good place to grow coffee? A. The elevation is high, the nights are cool and the days are humid. B. The elevation is low, the nights are warm and the days are dry. C. The elevation is high, the nights are warm and the days are dry. D. The elevation is low, the nights are cool and the days are dry. E. The elevation is high, the nights are warm and the days are humid. What specific temperature pattern have experts noted about the region where Kona coffee is grown in Hawaii? A. There has been no significant change but the bean production has dropped. B. The nights have warmed up, even though the days have cooled. C. There has been an increase in bean production with the change in climate. D. The nights have cooled even more so than before. E. There has been universally hot days all the way around.

http://www.econlib.org/library/Smith/smWN1.html#B.I,%20Ch.1,%20Of%20the%20Division%20of%20Labor What does Smith mean by division of labor, and … Read More...
Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n<D≪N. What is the sample space? If we are interested in knowing the number (count) of defective parts in the sample space, describe how, the concept of a random variable could help. Define a random variable Y and describe what values the random variable Y can take? What does it mean if we say Y=5? Q17. Suppose an experiment consists of tossing two fair coins. Let Y denote the number of heads appearing. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=1? What are the probabilities associated with each outcome? What is the sum of the probabilities associated with all possible values that Y can take? Q18. A lot, consisting of 100 fuses, is inspected by the following procedure. Five fuses are chosen at random and tested: if all 5 fuses pass the inspection, the lot is accepted. Suppose that the lot contains 20 defective fuses. What is the probability of accepting the lot? Define the random variable, its purpose, and the formula/concept that you would use. Q19. In a small pond there are 50 fish, 10 of which have been tagged. If a fisherman’s catch consists of 7 fish, selected at random and without replacement. Give an example of a random variable that can be defined if we are interested in knowing the number of tagged fish that are caught? What is the probability that exactly 2 tagged fish are caught? Define the random variable, its purpose, and the formula/concept that you would use. Applied to Quality Control Q20. My manufacturing firm makes 100 cars every day out of which 10 are defective; the quality control inspector tests drives 5 different cars. Based on the sample, the quality control inspector will make a generalization about the whole batch of 100 cars that I have on that day. Let d denote the number of defective cars in the sample What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective- call this P(d=0); (b) 20% of the cars are defective- call this P(d=1); (c) 40% of the cars are defective- call this P(d=2); (d) 60% of the cars are defective- call this P(d=3),(e) 80% of the cars are defective- call this P(d=4), and (f) 100% of the cars are defective- call this P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Q21. Assume that the quality control inspector is selecting 1 car at a time and the car that he tested is put back in the pool of possible cars that he can test (sample with replacement). Let d denote the number of defective cars in the sample (n) What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective, (b) 20% of the cars are defective, (c) 40% of the cars are defective, (d) 60% of the cars are defective, (e) 80% of the cars are defective, and (f) 100% of the cars are defective. Let’s call these P(d=0)….P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Interesting Problems Q22. A closet contains n pairs of shoes. If 2r shoes are chosen at random (2r<n), what is the probability that there will be no matching pair in the sample? Q23. In a draft lottery containing the 366 days of the leap year, what is the probability that the first 180 days drawn (without replacement) are evenly distributed among the 12 months? What is the probability that the first 30 days drawn contain none from September? Q25. You and I play a coin-tossing game. If the coin falls heads I score one, if tails, you score one. In the beginning, the score is zero. What is the probability that after 2n throws our scores are equal? What is the probability that after 2n+1 throws my score is three more than yours?

Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n

Que 1: verify for the cobb-Douglas Production function P (L, K) = 1.01L.75 K.25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by the double production?

Que 1: verify for the cobb-Douglas Production function P (L, K) = 1.01L.75 K.25 that the production will be doubled if both the amount of labor and the amount of capital are doubled. How much must you increase capital K to double production? How much must you increase labor by the double production?