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Economics 404 Winter 2014 -Midterm I




Question;Version 1Midterm I SolutionsEconomics 404 Winter 2014University of MichiganI. MULTIPLE CHOICE QUESTIONS:Each question is worth 5 points. The answers of these questions should go on the scantron sheet.1. If two events with positive probabilities are mutually exclusive, they cannot be independent.(a) True(b) False2. If Samantha has _ve blue socks and four green ones in her sock drawer, what is the probability that a randomly chosen group of three contains exactly two blue socks?(a) 0.476(b) 0.498(c) 0.255(d) 0.444(e) None of the above.3. If a basketball player misses free throws 10% of the time and this probability is constant every time he shoots, the probability that he will miss at least one of the next 3 shots is:(a) 0.729(b) 0.810(c) 0.370(d) 0.271(e) Not enough information to answer the question.4. Events A and B are independent and probability of A is 0.5. If the probability of the union of A and B is 0:9, what is the probability of event B?(a) 0.5(b) 0.8(c) 0.7(d) 0.6(e) None of the above5. Salma generated a 4 digit PIN code for her ATM card when she _rst got it. Ten years later, she realized that she still had a few bucks in that account and decided withdraw them. But she forgot the code besides the _rst and last digits. She remembers that it was 8??9. If numbers can take any valuebetween 0-9, but they cannot be repeated, what is the probability that she will guess the right code in her _rst attempt at the ATM machine?(a) 1=90(b) 1=56(c) 1=45(d) 1=72(e) The probability is zero, she'd better call the number on the card.6. Consider a random experiment of rolling 3 dice in a row (a six-sided fair one). Suppose that we are interested in the sum of the face values showing on the dice. What is the probability that this sum will be larger than or equal to 5?(a) 15/108(b) 212/216(c) 105/108(d) 180/126(e) None of the above.7. Turkish Lira (currency of Turkey) had a terrible last week. On Monday, it depreciated 4% against the dollar, on Tuesday depreciated 5%, on Wednesday depreciated 3%, and on Thurday it again depreciated 6%. Turkish Central Bank increased the interest rate Thursday night and was able to appreciate the currency 2% on Friday. What was the daily geometric average return (or loss) rate of Lira over lastweek?(Hint: You can think about the value of a foreign currency as a regular asset for the purposes of this question (like a stock). Depreciation means loss of value. Appreciation is the opposite.)(a) 0.0310(b) -0.0216(c) -0.0248(d) -0.0324(e) None of the above8. A health survey done by UHS asks students several questions: (i) What is your gender? \Male" or \Female"(ii) What is your current weight?(iii) What is your degree of satisfaction with your weight? Choose between\Not happy with it", \Somewhat happy with it", \Happy with it" What types of measurement scale do the answers to these questions have? (in the order i,ii,iii)(a) Ratio, Nominal, Ordinal(b) Ordinal, Ordinal, Nominal(c) Nominal, Ratio, Ordinal(d) Nominal, Interval, Ordinal(e) Nominal, Inteval, Nominal9. Suppose Econ 404 has two sections. Both sections have the same number of students. In section I, 15 percent of the students are female. In section II, 45 percent of students are female. What is the probability that a randomly selected student from Econ 404 class is in Section II given that he is male?(a) 0.555(b) 0.432(c) 0.52(d) 0.054(e) 0.39310. Yulia needs to take a trip tomorrow. She generally doesn't like to travel. If the weather is not good, she hates it even more. Her utility from the trip depends on the weather, which is unknown the day before. Her level of utility from the trip is a function of temperature during the trip (T), and isin this form:????5000 + 2T2. She received a weather forecast for tomorrow's temperature as the following: it is going to be 30 degrees with probability 0:2, 40 degrees with probability 0:5, and 50 degrees with probability 0:3. What is the expected value of tomorrow's temperature and expected valueof her utility from the trip?(a) 41 and 1638(b) 41 and -1540(c) 40 and -1638(d) 40 and 1540(e) 41 and -163811. True or False? (1) If we measure a sample correlation coe_cient in the vicinity of 0:9, this is indicative of a strong linear relationship between the two variables (2) If we measure a correlation coe_cient which is close to zero, this means there is no relationship between the two variables.(a) (1) True, (2) False(b) (1) False, (2) False(c) (1) False, (2) True(d) (1) True, (2) True(e) None of the above12. If Spud Webb (1986 NBA Dunk Contest winner) is one standard deviation below the mean in the male height distribution (which is symmetric and bell shaped), his height approximately corresponds to the following percentile of the distribution:(a) 5th(b) 32th(c) 68th(d) 34th(e) 16th13. Two balls are sequentially drawn without replacement (without putting the _rst pick back into the urn) from an urn containing a blue ball, a yellow ball and two red balls. Which of the following options describes the sample space?(a) B,Y, R(b) BB,YY,RR,BR,RB,BY,YB,YR,RY(c) BY,YB,BR,RB,YR,RY,RR(d) B,Y,R,BY,YB,BR,RB,YR,RY,RR(e) None of the above14. Suppose Steve runs a small boutique hotel with 3 rooms. He knows that 20% of all the reservations are cancelled on the day of the reservation. If he makes 4 reservations for a given day, what is the probability that at least one of the rooms in the hotel will be empty on that day? (hint: at least oneroom will be empty if less than 3 guests arrive) (a) 0.2160(b) 0.1646(c) 0.2096(d) 0.1808(e) None of the above15. Figure below represents the cumulative mass (or probability) function of a random variable. What is the probability that this random variable is strictly larger than 2?xF(x)1 2 30.30.61(a) 0.6(b) 0.7(c) 0.3(d) 0.4(e) None of the aboveII. PROBLEMS [29 points in total]SHOW YOUR WORK! Please show the intermediate steps of your calculations and the formulas you use, and explain your solutions. 1. This January was very cold in Ann Arbor and Michigan. To put things in perspective, below is the histogram and summary statistics of average January temperatures of the past 30 years in Ann Arbor. We have 30 observations in the data set and each observation measures (average) January temperature (in Fahrenheits) of each year. The observations are for years from 1984 to 2013.Mean Median Standard Deviation Min Max23.44 23.625 4.8 14.9 30.42Now, consider this January where average temperature was 8 degrees Fahrenheit. We will update this data set with the addition of January 2014.(a) Calculate the sample mean for the data set updated with January 2014 information. [3pts](b) When we update the data to include 2014, do you expect the sample median to be lower than the sample mean or higher? Justify. [4pts](c) As you know, metric system uses Celsius instead of Fahrenheits to measure temperatures. One can convert temperatures in Fahrenheits into Celsius by using the following (approximate) formula C _ 0:55F????17:7.Suppose we converted the all data values to Celsius. Calculate the mean, median and standard deviation of January temperatures in Ann Arbor measured in Celsius degrees between 1984 and 2013 (so 2014 update is not included). [7 pts]2. A stock market (\IST"-short for Imaginary Stock Market) contains stocks of (a large number of) companies from three di_erent sectors: auto (A), banking (B) and computers (C). Auto and banking _rms each compromise 30% of the stock listed on the market, and the remaining 40% of the stocks is of computer companies.Over the past year, prices of the stocks listed on IST either went up (U) or down (D). We know that among auto companies the proportion of stocks which have lost value is twice as much as the proportion of those which gained value. We also know that in the banking section, things were better.Given a company is in the banking sector, the probability of a randomly chosen stock having increased in value in the past year is twice as as much as the probability of its stock price having decreased. Lastly, the computer sector had the best year. For a computer company, the chances of its stock value going up is three times as much as its chances of going down. (Hint for the entire question: It is not necessary but might be helpful to create a joint probability table.)(a) What is the probability of a randomly chosen stock having gone down in value? [5pts](b) What is the probability that a randomly chosen stock is of an auto company given that its value has gone up? [5 pts](c) What is the prior probability of a randomly chosen stock to be of an auto company, and what is the posterior probability of the stock belonging to an auto company after the arrival of the information that the stock value went up? Write down in the probability notation. [2pts](d) Is the prior or posterior probability larger? Explain why in a brief sentence or two. [3pts]


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