STOR 151 SECTION 2 FINAL EXAMMAY 1 2010YOUR NAME:PID:Honor pledge: On my honor, I have neither given nor received unauthorized aid in this exam.SIGNATURE:Please write your answers in a blue book, except for the graph in Question 2 which is at theend of this exam. Please hand in this question paper as well as your blue book.This is an open book exam. Course text, personal notes and calculator are permitted. Youhave 3 hours to complete the test. Personal computers and cellphones are not allowed. If you haveany queries about the meaning of a question, ask the instructor for advice.SHOW ALL WORKING — even correct answers will not get full credit if it’s not clear howthey were obtained. Incorrect answers will gain substantial credit if the method of working issubstantially correct.Answer six of the eight questions. If you attempt more than six, all the answers will be gradedbut only the best six (complete questions) will count. Each question is worth a total of 20 pointsand the whole exam is worth 120 points (which will be rescaled to a maximum of 35 for gradingpurposes). Points for each individual part of a question are also given in square brackets.1. Two indep endent throws are made of an 8-sided die (exactly like a regular die, but there are8 faces instead of 6 — each of the numbers 1 through 8 is equally likely to come up). Thetotal of the two throws is noted, which could be any number between 2 and 16.Consider the eventsA: the first throw is one of 2, 3, 6 or 7B: the first throw is oddC: the sum of the two throws is 13 or greater(a) Are the events A and B independent? [6](b) Are the events A and C independent? [7](c) Are the events B and C independent? [7]2. A professor of English b elieves that students who watch an excessive amount of TV scorebadly in a reading comprehension test. She administers the test to 10 students and recordstheir scores (y). She also asks them how many hours of TV they watch per week (x). Theresults are:Student 1 2 3 4 5 6 7 8 9 10TV Hours x 0 1 8 6 8 3 1 20 2 1Score on Test y 64 59 53 78 60 58 71 20 63 391You can assume the following without checking: ¯x = 5, ¯y = 56.5, sx= 6.1, sy= 16.5, r =−0.65.(a) Find the coefficients, a and b, in the regression line ˆy = a + bx. [7](b) A scatterplot of the x and y values is shown at the back of this question paper. Drawin the regression line from part (a) on that scatterplot. [5](c) What is the predicted score for a student who watches 40 hours of TV per week? Doesthis seem a reasonable estimate? — explain why or why not. [5](d) Comment on the regression analysis. Are there any unusual features that might makeyou wary of the conclusion? [3]3. A casino game is played in the following way. The player has to pay $10 to enter the game.Then, a fair coin is tossed three times. If all three come up heads, the player collects a prizeof $50. When that happens, the coin is tossed a further seven times. If all seven of thosealso come up heads (so there have been ten heads in a row), the player receives an additional$2,000.(a) Write down a table of the possible payouts and their respective probabilities. [5](b) What is the mean payout to the player on a single play of the game? [5](c) An observer writes down the actual payout for each play over a long sequence of plays.The standard deviation of these payouts is calculated to be ab out $66. Based on thisinformation, if ¯x denotes the mean payout after n plays of the game, what are the meanand standard deviation of ¯x? [3](d) The manager of the casino reckons that the game is played 900 times in the course of asingle evening. What is the probability that by the end of the evening, the casino hasmade a profit? [7]4. A medical research organization would like to conduct a survey to estimate the proportion ofpeople with diabetes in a native American tribe. A random sample of tribe members will beselected and tested for diabetes. The researcher would like to construct a confidence intervalfor the proportion of tribe members who have diabetes, with confidence coefficient 99% anda margin of error no more than .028.(a) What sample size is needed if the researcher makes no prior assumption about the truerate of diabetes in the population? [6](b) It is known that the overall proportion of people with diabetes in the American popula-tion is about .08. Based on this information, the researcher is willing to assume that therate of diabetes in the native American tribe is not more than .15. If the sample size isrecalculated under this assumption, what sample size is then needed? [6](c) In fact, the test proceeds with a sample of size 921, among whom 166 are found tohave diabetes. Find a 99% confidence interval for the true proportion of tribe memberswho have diabetes. Based on this confidence interval, do you think the assumption theresearcher made in (b) was justified? [8]25. A brand of beer advertises that it contains 5% alcohol. In order to test that claim, a consumerorganization collects 5 bottles of beer and measures their alcohol content. The results for the5 bottles are: 4.2%, 4.6%, 4.6%, 4.4%, 4.2%.(a) Assume the 5 bottles of beer are a random sample from the population of all bottles ofbeer of this particular brand. Construct a 95% confidence interval for the mean alcoholcontent of the brand. [8](b) Does this test provide statistically significant evidence that the true mean alcohol contentin the beer is less than 5%? Summarize the evidence in terms of a P-value (as closely asyou are able to determine it), and explain how you interpret the conclusion. [8](c) What are the assumptions underyling this statistical procedure? Do you believe theyare satisfied in this example? [4]6. A recent New York Times/CBS News Poll interviewed 881 adults who identified themselvesas supporters of the Tea Party, and a separate sample of 1580 from the general population.Among the Tea Party supporters, 41% stated that they believed President Obama was bornin the United States. Among the general population sample, 58% stated that they believedPresident Obama was born in the United States.(a) Find a 90% confidence interval for the difference between the proportion of Tea Partysupporters who believe President Obama was born in the United States and the pro-portion of the general population who believe President Obama was born in the UnitedStates. [10](b) Do you believe the two proportions are equal? Formulate this as a hypothesis testingproblem, carry out the test, and state your