STATISTICS FOR DECISION MAKING Dr. Azar Abizada HOMEWORK 4 Due Date: Wednesday, March 11th class time Problem 1 (26 points): Be careful, each part of this question is independent of one another. a) (7 points) Suppose that the probability is 0.2 that the value of the U.S. dollar will rise against the Chinese yuan over any given week and that the outcome in one week is independent of that in any other week. What is the probability that the value of U.S. dollar will rise against Chinese yuan at least twice over a period of 7 weeks? b) (6 points) A customer service center in India receives, on average 5.4 telephone calls per minute. If the distribution of calls is Poisson, what is the probability of receiving at least 3 calls during next two minutes? c) (6 points) X and Y are random variables with finite variance. Which one is larger: Var(X+Y) or Var(X-Y)? d) (7 points) Assume that each birth is an independent event with an equal chance of being male or female. If you have four children, they may all be of one sex, there may be three of one sex and one of the other sex, or there may be two of each. Which is most likely? Problem 2 (20 points): There are two locations in town (north and south) under consideration for a new restaurant, but only one location will actually become available. If it is in the north, the restaurant stands a 90% chance of surviving its first year. However, if it is built in the south, its chances of survival are only 65%. It is estimated that the chances of the northern location being available are 40%. a) (10 points) Find the probability that the restaurant will survive its first year. b) (10 points) Find the probability that the restaurant is in the south given that it is successful. Problem 3 (25 points): Soft Drink Company Juicy has two small stores and one large store in the city. Daily sales of a drink in a large store are normally distributed with a mean of 200 and a standard deviation of 20. Daily sales of a drink in the small store are also normally distributed with a mean of 120 and standard deviation of 10. Sales levels in two types of stores have a correlation of 0.80. The selling price per drink is $10. The total production costs in all stores per day are $4000 (this is the sum of the costs of three stores). What is the probability that company makes a loss on any day? Problem 4 (29 points): An experimental bumper was designed to reduce damage in low speed collisions. This bumper was installed on an experimental group of vans in a large fleet, but not on a control group. At the end of a trial period, accident data showed 12 repair incidents (a \"repair incident\" is a repair invoice) for the experimental vehicles and 9 repair incidents for the control group vehicles. The sample mean value for repair incidents for the experimental group was $1,101.42. For the control group the sample mean was $1,766.11. The sample standard deviations were sE = $696.2 for the experimental group and sC = $837.62 for the control group. Suppose these two populations are normally distributed with equal variances. a) (5 points) Construct 99% confidence interval for population average value of repair incidents for the experimental group. b) (5 points) Construct 98% confidence interval for population average value of repair incidents for the control group. c) (6 points) Construct 95% confidence interval for population variance of value of repair incidents for the experimental group. d) (6 points) Construct 90% confidence interval for population variance of value of repair incidents for the control group. e) (7 points) Construct 95% confidence interval for true difference between average values of repair incidents for two groups. Econ 503 Assignment 3 Due date: April 13, 2015 by 9:00am Questions 1. Suppose that we can represent the aggregate production for Azerbaijan by a function of CobbDouglas form: Y = AK 0.3 L0.7 , with A = 2. Note that this function has the same form with the function we used in class with = 0.3, and the technology parameter A = 2. Further, assume that the saving rate, s, is equal to 0.4, the population growth rate, n, is 0.01, and capital depreciates at a rate = 0.09. (a) Write the production function in per-capita terms, i.e., y = f (k). (b) Solve for steady state level of capital per-capita, k . (Hint: Write down law of motion for k and set the change in capital equal to zero.) (c) Assume that A goes down to 1. Calculate the new steady state value for capital per-capita, k . (d) Compare the consumption levels in part (b) and (c). 2. In this question, you will numerically verify the result you obtained in question (1) part (c). Download the Excel le \"Asgnmt3.xls\" from PowerCampus. Go to the excel sheet called \"SteadyState\". You will nd an empty table in that sheet. The very rst row depicts the parameters of the Solow model. For example, s is set to 0.4, and A is set to 1, as in the rst question. The rst column is an index of the current period. It starts with zero and goes all the way to 150. The second column is the capital per-capita level in the current period. For period=0, this is set sot10. Your task is to ll the rest of the table. The rest of the columns are as follows: output per-capita, savings and investment, consumption, depreciation and population growth, change in capital. 3. For this question, assume that A = 1, and the rest of the parameters are as in question 1. (a) Calculate golden rule level of capital per-capita, kg . (b) In this part, you will verify your answer in part (a) of this question. In the Excel le \"Asgnmt3.xls\\fs= k period 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 y 10.00000 0.4 n= sy=i c=y-i 0.01 d= (d+n)k 0.09 A= Dk 1 a= 0.3 n= s k* 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.01 d= y* 0.09 A= c* 1 a= 0.3 Econ 503 Assignment 5 Due date: April 27, 2015 by 9:00am Questions 1. Recall the equation from the quantity theory of money, M V = P Y . We can use this relationship to describe the aggregate demand. (a) For xed values of M and V , draw on a well-labeled graph the relationship between P and Y . (b) According to the classical view, total output in the economy is determined by that country's resources and technology. Under this view, how does the aggregate supply look like? Plot an aggregate supply curve that is consistent with the classical view. Label it as CAS. (c) Suppose that the central bank decreases the money supply, M . What happens to output under classical view? What happens to the general price level? Is the model consistent with the predictions of the quantity theory of money regarding the relationship between ination and money? Briey comment on that. 2. Right after the Great Depression, Keynes challenged the classical view on the behavior of the general price level. He argued that prices do not immediately adjust in response to changes in aggregate demand. (a) Which real life observation(s) motivated Keynesian ideas? Please briey explain why prices can be sticky in the short-run. (b) Under the Keynesian view, how does the aggregate supply look like? On a new graph, plot an aggregate supply curve that is consistent with the Keynesian view. Label it as KAS. Also, draw an aggregate demand curve as in question (1). (c) Suppose that the central bank decreases the money supply, M . What happens to output under Keynesian view? What happens to the general price level? What is the shortrun relationship between ination and unemployment? (Hint: We will later depict this relationship by a curve called Philips curve.) 1 Statistics for Decision Making Dr. Azar Abizada Homework 5 /Practice Final Exam Deadline: April 27th, 2015, class time Problem 1: An experimental bumper was designed to reduce damage in low speed collisions. This bumper was installed on an experimental group of vans in a large fleet, but not on a control group. At the end of a trial period, accident data showed 12 repair incidents (a \"repair incident\" is a repair invoice) for the experimental vehicles and 9 repair incidents for the control group vehicles. The sample mean value for repair incidents for the experimental group was $1,101.42. For the control group the sample mean was $1,766.11. The sample standard deviations were sE = $696.2 for the experimental group and sC = $837.62 for the control group. Can we conclude that the design of the new bumper was successful? Problem 2: Each day, a fast-food chain tests that the average weight of its \"two-pounders\" is at most 32 ounces. The alternative hypothesis is that the average weight is more than 32 ounces, indicating that new processing procedures are needed. The weights of two-pounders can be assumed to be normally distributed, with a standard deviation of 3 ounces. The decision rule adopted is to reject the null hypothesis if the sample mean weight is more than 33 ounces. a) If a random sample of n=49 two-pounders are selected, what is the probability of a Type I error, using this decision rule? b) Suppose that the true mean weight is 34 ounces. If random samples of 25 two-pounders are selected, what is the probability of a Type II error, using this decision rule? Problem 3: An ad agency has developed a TV ad for the introduction of a toothpaste. The objective of the ad is to create brand awareness. The effectiveness of the ad is measured by aided- and unaided-recall scores. 200 respondents who claimed to have watched the TV show in which the ad was aired the night before have been contacted by telephone in 20 cities. In order for the ad to be successful, the percentage of unaided calls must be above 18 percent, which is the category norm for a TV commercial for the product class. Suppose out of 200 respondents 46 of them were able to recall the commercial without any prompting (unaided recall). Based on the information contained in the data, can we conclude that the ad was successful? 1 Problem 4: A corporation administers an aptitude test to all new sales representatives. Management is interested in the extent to which this test is able to predict their eventual success. The accompanying table records average weekly sales (in thousands of dollars) and aptitude test scores for a random sample of eight representatives: Sales 10 12 28 24 18 16 15 12 Score 55 60 85 75 80 85 65 60 Regression Statistics Multiple R 0.774781 R Square 0.600285 Adjusted R Square 0.533666 Standard Error 4.279244 Observations 8 ANOVA Regression Residual Total Intercept X Variable 1 df 1 6 7 Significance SS MS F F 165.0034 165.0034 9.010709 0.023953 109.8716 18.31193 274.875 Standard Coefficients Error t Stat P-value Lower 95% -11.5046 9.57453 -1.20158 0.274797 -34.9326 0.401835 0.133865 3.001784 0.023953 0.074278 a) Estimate the linear regression of weekly sales on aptitude test scores. Interpret the estimated slope of the regression line. b) Find and interpret the coefficient of determination. c) Test the null hypothesis H0: 1 = 0 against the alternative at 5% level of significance. Interpret the result. 2 Problem 5: The midterm grades of the students in Statistics course follows a normal distribution with mean 80 and a standard deviation of 8. Professor gives grade A if a student scores above 92, grade B if a student scores between 76 and 92, grade C if a student score between 60 and 76 and grade F if a student scores below 54. a) What percent of the students get grade A? b) What percent of the students get grade C? c) If a student passed the course, what is the probability that he passed with grade B? Problem 6: An auditor checked a random sample of 16 of the accounts payable of a company. The sample mean was $131 with standard deviation $25. Assume that the population distribution is normal. a) Construct 95% confidence interval for population mean and variance of accounts payable. b) CFO gets notified when variance of accounts payable is above 800. Is there enough reason to notify the CFO? Problem 7: Consolidated Power, a large electric utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is 60 F or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of 100 temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly, then the plant must be shut down and appropriate actions must be taken to correct the problem. a) Consolidated Power wishes to set up a hypothesis test so that the power plant will be shut down when the null hypothesis is rejected. Set up the null hypothesis H0 and the alternative that should be used. Suppose that Consolidated Power decides to use 5% significance level. Suppose also that a random sample of 100 temperature readings is obtained. If the sample mean of these readings is 60.482 test the null hypothesis and determine whether the power plant should be shut down and the cooling system repaired. Assume =2. b) Assuming that the actual population mean water temperature is 60.6 calculate Type II error of the test. 3 Econ 503 Assignment 3 Due date: April 13, 2015 by 9:00am Questions 1. Suppose that we can represent the aggregate production for Azerbaijan by a function of CobbDouglas form: Y = AK 0.3 L0.7 , with A = 2. Note that this function has the same form with the function we used in class with = 0.3, and the technology parameter A = 2. Further, assume that the saving rate, s, is equal to 0.4, the population growth rate, n, is 0.01, and capital depreciates at a rate = 0.09. (a) Write the production function in per-capita terms, i.e., y = f (k). (b) Solve for steady state level of capital per-capita, k . (Hint: Write down law of motion for k and set the change in capital equal to zero.) (c) Assume that A goes down to 1. Calculate the new steady state value for capital per-capita, k . (d) Compare the consumption levels in part (b) and (c). 2. In this question, you will numerically verify the result you obtained in question (1) part (c). Download the Excel le \"Asgnmt3.xls\" from PowerCampus. Go to the excel sheet called \"SteadyState\". You will nd an empty table in that sheet. The very rst row depicts the parameters of the Solow model. For example, s is set to 0.4, and A is set to 1, as in the rst question. The rst column is an index of the current period. It starts with zero and goes all the way to 150. The second column is the capital per-capita level in the current period. For period=0, this is set sot10. Your task is to ll the rest of the table. The rest of the columns are as follows: output per-capita, savings and investment, consumption, depreciation and population growth, change in capital. 3. For this question, assume that A = 1, and the rest of the parameters are as in question 1. (a) Calculate golden rule level of capital per-capita, kg . (b) In this part, you will verify your answer in part (a) of this question. In the Excel le \"Asgnmt3.xls\\fs= k period 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 y 10.00000 0.4 n= sy=i c=y-i 0.01 d= (d+n)k 0.09 A= Dk 1 a= 0.3 n= s k* 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.01 d= y* 0.09 A= c* 1 a= 0.3 Econ 503 Assignment 5 Due date: April 27, 2015 by 9:00am Questions 1. Recall the equation from the quantity theory of money, M V = P Y . We can use this relationship to describe the aggregate demand. (a) For xed values of M and V , draw on a well-labeled graph the relationship between P and Y . (b) According to the classical view, total output in the economy is determined by that country's resources and technology. Under this view, how does the aggregate supply look like? Plot an aggregate supply curve that is consistent with the classical view. Label it as CAS. (c) Suppose that the central bank decreases the money supply, M . What happens to output under classical view? What happens to the general price level? Is the model consistent with the predictions of the quantity theory of money regarding the relationship between ination and money? Briey comment on that. 2. Right after the Great Depression, Keynes challenged the classical view on the behavior of the general price level. He argued that prices do not immediately adjust in response to changes in aggregate demand. (a) Which real life observation(s) motivated Keynesian ideas? Please briey explain why prices can be sticky in the short-run. (b) Under the Keynesian view, how does the aggregate supply look like? On a new graph, plot an aggregate supply curve that is consistent with the Keynesian view. Label it as KAS. Also, draw an aggregate demand curve as in question (1). (c) Suppose that the central bank decreases the money supply, M . What happens to output under Keynesian view? What happens to the general price level? What is the shortrun relationship between ination and unemployment? (Hint: We will later depict this relationship by a curve called Philips curve.) 1 Statistics for Decision Making Dr. Azar Abizada Homework 5 /Practice Final Exam Deadline: April 27th, 2015, class time Problem 1: An experimental bumper was designed to reduce damage in low speed collisions. This bumper was installed on an experimental group of vans in a large fleet, but not on a control group. At the end of a trial period, accident data showed 12 repair incidents (a \"repair incident\" is a repair invoice) for the experimental vehicles and 9 repair incidents for the control group vehicles. The sample mean value for repair incidents for the experimental group was $1,101.42. For the control group the sample mean was $1,766.11. The sample standard deviations were sE = $696.2 for the experimental group and sC = $837.62 for the control group. Can we conclude that the design of the new bumper was successful? Problem 2: Each day, a fast-food chain tests that the average weight of its \"two-pounders\" is at most 32 ounces. The alternative hypothesis is that the average weight is more than 32 ounces, indicating that new processing procedures are needed. The weights of two-pounders can be assumed to be normally distributed, with a standard deviation of 3 ounces. The decision rule adopted is to reject the null hypothesis if the sample mean weight is more than 33 ounces. a) If a random sample of n=49 two-pounders are selected, what is the probability of a Type I error, using this decision rule? b) Suppose that the true mean weight is 34 ounces. If random samples of 25 two-pounders are selected, what is the probability of a Type II error, using this decision rule? Problem 3: An ad agency has developed a TV ad for the introduction of a toothpaste. The objective of the ad is to create brand awareness. The effectiveness of the ad is measured by aided- and unaided-recall scores. 200 respondents who claimed to have watched the TV show in which the ad was aired the night before have been contacted by telephone in 20 cities. In order for the ad to be successful, the percentage of unaided calls must be above 18 percent, which is the category norm for a TV commercial for the product class. Suppose out of 200 respondents 46 of them were able to recall the commercial without any prompting (unaided recall). Based on the information contained in the data, can we conclude that the ad was successful? 1 Problem 4: A corporation administers an aptitude test to all new sales representatives. Management is interested in the extent to which this test is able to predict their eventual success. The accompanying table records average weekly sales (in thousands of dollars) and aptitude test scores for a random sample of eight representatives: Sales 10 12 28 24 18 16 15 12 Score 55 60 85 75 80 85 65 60 Regression Statistics Multiple R 0.774781 R Square 0.600285 Adjusted R Square 0.533666 Standard Error 4.279244 Observations 8 ANOVA Regression Residual Total Intercept X Variable 1 df 1 6 7 Significance SS MS F F 165.0034 165.0034 9.010709 0.023953 109.8716 18.31193 274.875 Standard Coefficients Error t Stat P-value Lower 95% -11.5046 9.57453 -1.20158 0.274797 -34.9326 0.401835 0.133865 3.001784 0.023953 0.074278 a) Estimate the linear regression of weekly sales on aptitude test scores. Interpret the estimated slope of the regression line. b) Find and interpret the coefficient of determination. c) Test the null hypothesis H0: 1 = 0 against the alternative at 5% level of significance. Interpret the result. 2 Problem 5: The midterm grades of the students in Statistics course follows a normal distribution with mean 80 and a standard deviation of 8. Professor gives grade A if a student scores above 92, grade B if a student scores between 76 and 92, grade C if a student score between 60 and 76 and grade F if a student scores below 54. a) What percent of the students get grade A? b) What percent of the students get grade C? c) If a student passed the course, what is the probability that he passed with grade B? Problem 6: An auditor checked a random sample of 16 of the accounts payable of a company. The sample mean was $131 with standard deviation $25. Assume that the population distribution is normal. a) Construct 95% confidence interval for population mean and variance of accounts payable. b) CFO gets notified when variance of accounts payable is above 800. Is there enough reason to notify the CFO? Problem 7: Consolidated Power, a large electric utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is 60 F or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of 100 temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly, then the plant must be shut down and appropriate actions must be taken to correct the problem. a) Consolidated Power wishes to set up a hypothesis test so that the power plant will be shut down when the null hypothesis is rejected. Set up the null hypothesis H0 and the alternative that should be used. Suppose that Consolidated Power decides to use 5% significance level. Suppose also that a random sample of 100 temperature readings is obtained. If the sample mean of these readings is 60.482 test the null hypothesis and determine whether the power plant should be shut down and the cooling system repaired. Assume =2. b) Assuming that the actual population mean water temperature is 60.6 calculate Type II error of the test. 3