Department of Economics Econ 395 Assignment 1

Department of EconomicsEcon 395Assignment 1Spring 2016Instructions:• This assignment is due on Tuesday, May 24, 2016 at the beginning of the class.• Late assignments, assignments sent by e-mail, placed in my mailbox or placed under myoffice door will not be accepted.• Students may work in small groups of 2 students on assignments, and a group file is to behanded in. Students in the group will receive the same mark for the assignment.• Your assignment should be done in a professional manner. Part marks will bededucted for leaving out formula, not showing calculations and all the steps, not writingsentence answers to all questions, not clearly labelling the question and the parts of thequestions, not including a cover page with your name(s), not stapling the assignment, andnot numbering the pages.1.Express each of the following sums in summation notation and then compute wherepossible.Let X take the values ! = −4, ! = −2, ! = 0, ! = 4, ! = 4 and Y take the values! = −1, ! = −0.5, ! = 0, ! = 1, ! = 1.5.a)b)c)d)e)f)! + ! + ! + ! + !! + ! + !(! − 2! ) + (! − 2! ) + (! − 2! ) + (! − 2! ) + (! − 2! )(! ! ) + (! ! ) + (! ! )(3! ! ) + (3! ! ) + (3! ! )(! ! ! ) + (! ! ! ) + (! ! ! )!!!!(!− ) ! − =!!!! !! −2.Show that3.An economist for the federal government is attempting to produce a better measure ofpoverty than is currently in use. To help acquire information, she recorded the annualhousehold income, in thousands of dollars, and the amount of money spent on foodduring one week for a random sample of households. Below are four equations that wereSpring 2016 Econ 395 Assignment 1 continuedPage 2estimated by the economist with the data where x is the annual income (in $1,000) and yis the weekly food expenditure for the household.= 153.90 + 1.96ln = 3.91 + 0.41 ln= −172.34 + 108.81ln ()ln = 5.15 + 0.007a) In Excel, in one graph, plot each of the functions for x = 40 to x = 80.b) Calculate the slope of each function at x = 60. State the interpretation of the slope.(Hint: use table A.2)c) Calculate the elasticity of each function at x = 60 and give its interpretation.4.Let X be a discrete random variable that is the value shown on a single roll of a fair die.a) What is the probability the X = 1 or X = 3 or X = 5?b) What is the expected value of X? What is the expected value of X2?c) Find the variance of X.d) Find the expected value and variance of W if W = g(X) = 8X – 4.e) Now, assume the die has been weighted so that P(X=1) = 1/3, P(X=2) = 1/3, and theP(X=3) = P(X=4) = P(X=5) = P(X=6) = 1/12. Find the answers to parts b) and c)under the new assumption.5.After watching a number of children playing games in a video arcade, a statisticspractitioner estimated the following probability distribution of X, the number of gamesper visit.x12345f(x).05.20.30.25.20a) What is the probability that a child will play at most three games?b) What is the mean and variance of the number of games played?c) Suppose that each game costs $2 to play. Use the laws of expected value andvariance to determine the expected value and variance of the revenue (the amount ofmoney) the arcade receives per child.d) Determine the probability distribution of the revenue the arcade receives per child.e) Use the probability distribution you created in part d) to calculate the mean andvariance of the revenue the arcade receives per child.f) Are your answers in part c) & part e) identical?6.The decision about where to build a new plant is a major one for most companies. One ofthe factors often considered is the education level of the location’s residents. Censusinformation may be useful in this regard. After analyzing a recent census, a companyproduced the following joint probabilities:Spring 2016 Econ 395 Assignment 1 continuedEducationNot a high school graduateHigh school graduateSome college, no degreeAssociate’s degreeBachelor’s degreeAdvanced degreePage 3NorthwestMidwestSouthWest0.03010.03180.06830.03590.07110.08430.11740.06080.02620.04100.06050.04560.01430.01800.02480.01810.03500.03680.05590.04180.01900.01840.02690.0180a) What is the probability of a randomly selected individual having an associate’sdegree and living in the Midwest?b) Determine the probability that a person living in the South has a bachelor’s degree.c) Find the probability that a high school graduate lives in the West.d) What is the probability that a randomly selected individual lives in the Northwest?e) Are education and location independent?7.How does level of affluence affect health care? To address one dimension of the problema group of heart attack victims was selected. Each was categorized as a low-, medium-,or high-income earner. Each was also categorized as having survived or died. Ademographer notes that in our society 25% fall into the low-income group, 55% into themedium-income group and 20% are in the high-income group. Furthermore, an analysisof heart attack victims reveals that 12% of low-income people, 9% of medium-incomepeople and 7% of high-income people die of heart attacks.a) What is the probability of dying from a heart attack?b) Find the probability that a survivor of a heart attack is in the high-income group.8.Let X be a continuous random variable with probability density function given by= −0.5 + 1, 0 ≤ ≤ 2a) Graph the density function .b) Find the total area beneath for 0 ≤ ≤ 2;c) What is the probability that 0.5 ≤ ≤ 1.5?9.The final marks in an economics course are normally distributed with a mean of 70 and astandard deviation of 8. The professor must convert all the marks to letter grades. Shedecides that she wants 15% A’s, 38% B’s, 35% C’s, 10% D’s, and 2% F’s. Determinethe cutoffs (what the actual marks are) for each letter grade.