ECON 210 - Economic Statistics Summer 2016 Final Examination Things to Remember: This is a take-home Final Examination being administered on Thursday, July 21, 2016 and due on Tuesday, July 26, 2016 at 12:00 PM (noon). Each question is worth 10 points for a total of 100 points. If a question has more than one part, the points are equally distributed amongst the parts. Partial Credit is available, so turn in all your work! You are not allowed to consult sources on the internet in order to solve any of these problems unless explicitly asked to do so. You are not allowed to work with your classmates. All exams must reflect an individual effort. You are, however, free to consult your class notes and the textbook and use any information therein to answer these questions. You can turn your exams in to me in person before the deadline by setting up an appointment with me beforehand, Saturday and Sunday excluded. You are also free to email your exams to me. If you choose to do so, I strongly prefer that the exams are type-written. In case you choose to send me a scanned copy of a hand-written examination, please make sure all your answers are legible on the scan. If I can't read it, it will not get any points. Some of the questions involve some elementary data analysis using Excel. The data that each question requires have been uploaded on D2L and are marked with the question numbers. Please make sure you use the correct data. If it isn't mentioned explicitly, you must report p-values for every hypothesis test you perform. The exam begins on the next page. Best of Luck! 1 1) Suppose you place a bet with your friend that two people in a room filled with a certain number of people will have the same birthday (just the day of the year, they needn't be the same age). How many people do you need in this room in order to make the bet favorable to you, i.e. you have at least a 50% chance of winning? Ignore leap years in your calculation. [Hint: Use the knowledge that P(at least one) + P(none) = 1, i.e. P(at least one pair in a room of n people have same birthday) + P(no pair in a room of n people have same birthday) = 1. Hence find what n needs to be such that P(no pair in a room of n people have same birthday) < 0.5. Think Permutations.] 2) I have in front of me a diagram of a n sided polygon. I draw a line connecting two vertices of this polygon. What is the probability that the line I've drawn is a diagonal? [Hint: Think Combinations.] 3) A middle aged couple are out to buy life insurance and perform the required medical tests. The underwriters of the insurance policy estimate that the probability that the man will be alive 25 years from now is 0.3 and the probability that the lady will be alive 25 years from now is 0.4. What is the probability that: (i) neither will be alive 25 years from now? (ii) at least one will be alive 25 years from now? 4) According to the CDC, \"Compared to nonsmokers, men who smoke are about 23 times more likely to develop lung cancer and women who smoke are about 13 times more likely.\" The CDC has also found that 17.9% of women smoke. If you learn that a woman has been diagnosed with lung cancer, and you know nothing else about her, what is the probability that she is a smoker? [Hint: You clearly have very limited information. Try to find an algebraic expression for this probability and then use the percentage of women who smoke to arrive at the answer.] 5) Find the probability distribution of the number of failures preceding the first success in an infinite series of independent trials of an experiment with a constant success probability p for each trial. Also, find the expected value of the number of failures preceding the first success. You will need the following information: \"The infinite sum of the elements of an Arithmetico-Geometric Series, i.e. a series of the form {1, 2x, 3x2 , 4x3 , ....}, is given by S = 1/(1 x)2 .\" [Hint: go up to the case with a success on the fifth trial, you should be able to see a pattern emerging. By the way, if you've done this correctly, then you've figured out the expression for a Geometric Distribution with parameter p !] 2 6) Physicists say that particles of an ideal gas constantly move back and forth along a sealed long tube, each with a velocity V (in cm/sec), at any given moment. We are also told that theoretically, these velocities follow the Maxwell-Boltzmann Distribution with parameter a = 2 (specific qto the gas). The mean and variance of the Maxwell-Boltzmann Distribution are given by = 2a 2 and 2 = a2 (38) . Suppose there are a 100 particles in the tube. (i) What is the distribution of the average velocity of these particles? (ii) What is the probability that the average velocity exceeds 3.45 cm/sec? (iii) Find the 99% Confidence Intervals for the average velocity of the particles. 7) The dataset for Question 7 lists Barry Bonds' batting statistics by season for his entire career. Barry Bonds was alleged to have used performance enhancing drugs starting 1998 and we wish to test if there may be some truth to these allegations. Consider the data coming from seasons after 1998 to be a sample. Calculate the mean values for all the statistics pre- and post-1998. Treat the pre-1998 mean as the population mean and the post-1998 mean as the sample mean. Calculate the standard deviation post-1998 and treat that as the sample standard deviation. Construct tests of 5% significance using the measures HR, BA, SLG, OBP and OPS that test if there is any evidence to support the use of performance enhancing drugs by Mr. Bonds. Mention if you're using a onetailed or a two-tailed test. [Hint: Keep track of the sample size and use the appropriate test.] 8) A silicon-valley marketing guru employed by Apple Inc. takes a hit of LSD and hypothesizes that if the iPhone 6 were placed under a blue light (particularly that of Hex Number 8470ff) it will induce an unsuspecting customer to purchase it more often. Apple Inc., always trusting in the power of LSD to generate deeper insight into selling stuff (and life in general), decides to act on this immediately. So, it selects a random sample of 225 stores where the iPhone 6 are placed under said blue light. It also deploys an army of Steve Jobs fanboys to collect data and calculate the average sales volume from stores with the new lighting. The data is as follows: In stores with the lighting, the average sales volume is $328,754 with a standard deviation of $227,000. The average sales volume per store, from previous data, was $300,000. Does this new sales strategy impact the buying habit of the customer in any way? Set up a hypothesis test of significance value 5% to test this. [Hint: If you haven't figured this out already, I am an Android person.] 9) Consider the dataset for Question 9, which produces a random sampling of the political opinions of individuals scoured from the dark corners of the web. We want to know if this particular sample is, on average, more disdainful of Hilary Clinton or Donald Trump. Is the data paired or un-paired? Set up a test of = 0.05 to figure out if the people hate Hilary or Donald more. [Hint: Keep track 3 of the sample size and use the appropriate test.] 10) The dataset for Question 10 lists the forty yard dash times by Wide Receivers and Cornerbacks in the NFL Combines in two separate sheets. Test the hypothesis that Wide Receivers are slower than Cornerbacks at 5% level of significance. [Hint: Is this data paired?] 4 \fBarry Bonds' Season by Season Batting Stats Year Age 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 Tm 21 PIT 22 PIT 23 PIT 24 PIT 25 PIT 26 PIT 27 PIT 28 SFG 29 SFG 30 SFG 31 SFG 32 SFG 33 SFG 34 SFG 35 SFG 36 SFG 37 SFG 38 SFG 39 SFG 40 SFG 41 SFG 42 SFG Lg NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL NL HR BA 16 25 24 19 33 25 34 46 37 33 42 40 37 34 49 73 46 45 45 5 26 28 OBP 0.223 0.261 0.283 0.248 0.301 0.292 0.311 0.336 0.312 0.294 0.308 0.291 0.303 0.262 0.306 0.328 0.37 0.341 0.362 0.286 0.27 0.276 SLG 0.33 0.329 0.368 0.351 0.406 0.41 0.456 0.458 0.426 0.431 0.461 0.446 0.438 0.389 0.44 0.515 0.582 0.529 0.609 0.404 0.454 0.48 OPS 0.416 0.492 0.491 0.426 0.565 0.514 0.624 0.677 0.647 0.577 0.615 0.585 0.609 0.617 0.688 0.863 0.799 0.749 0.812 0.667 0.545 0.565 0.746 0.821 0.859 0.777 0.97 0.924 1.08 1.136 1.073 1.009 1.076 1.031 1.047 1.006 1.127 1.379 1.381 1.278 1.422 1.071 0.999 1.045 \f