STOR 151 SECTION 2 FINAL EXAM MAY 1 2010 PID Honor pledge On my honor I have neither given nor received unauthorized aid in this exam YOUR NAME SIGNATURE Please write your answers in a blue book except for the graph in Question 2 which is at the end 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 You have 3 hours to complete the test Personal computers and cellphones are not allowed If you have any 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 how Incorrect answers will gain substantial credit if the method of working is they were obtained substantially correct Answer six of the eight questions If you attempt more than six all the answers will be graded but only the best six complete questions will count Each question is worth a total of 20 points and the whole exam is worth 120 points which will be rescaled to a maximum of 35 for grading purposes Points for each individual part of a question are also given in square brackets 1 Two independent throws are made of an 8 sided die exactly like a regular die but there are 8 faces instead of 6 each of the numbers 1 through 8 is equally likely to come up The total of the two throws is noted which could be any number between 2 and 16 Consider the events A the rst throw is one of 2 3 6 or 7 B the rst throw is odd C 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 believes that students who watch an excessive amount of TV score badly in a reading comprehension test She administers the test to 10 students and records their scores y She also asks them how many hours of TV they watch per week x The results are Student TV Hours x Score on Test y 1 0 64 2 1 59 4 6 78 5 8 60 6 3 58 7 1 71 8 20 20 9 2 63 10 1 39 3 8 53 1 You can assume the following without checking x 5 y 56 5 sx 6 1 sy 16 5 r 0 65 a Find the coe cients a and b in the regression line y a bx 7 b A scatterplot of the x and y values is shown at the back of this question paper Draw in 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 Does this seem a reasonable estimate explain why or why not 5 d Comment on the regression analysis Are there any unusual features that might make you 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 prize of 50 When that happens the coin is tossed a further seven times If all seven of those also 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 about 66 Based on this information if x denotes the mean payout after n plays of the game what are the mean and standard deviation of x 3 d The manager of the casino reckons that the game is played 900 times in the course of a single evening What is the probability that by the end of the evening the casino has made a pro t 7 4 A medical research organization would like to conduct a survey to estimate the proportion of people with diabetes in a native American tribe A random sample of tribe members will be selected and tested for diabetes The researcher would like to construct a con dence interval for the proportion of tribe members who have diabetes with con dence coe cient 99 and a margin of error no more than 028 a What sample size is needed if the researcher makes no prior assumption about the true rate 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 the rate of diabetes in the native American tribe is not more than 15 If the sample size is recalculated 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 to have diabetes Find a 99 con dence interval for the true proportion of tribe members who have diabetes Based on this con dence interval do you think the assumption the researcher made in b was justi ed 8 2 5 A brand of beer advertises that it contains 5 alcohol In order to test that claim a consumer organization collects 5 bottles of beer and measures their alcohol content The results for the 5 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 of beer of this particular brand Construct a 95 con dence interval for the mean alcohol content of the brand 8 b Does this test provide statistically signi cant evidence that the true mean alcohol content in the beer is less than 5 Summarize the evidence in terms of a P value as closely as you 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 they are satis ed in this example 4 6 A recent New York Times CBS News Poll interviewed 881 adults who identi ed themselves as 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 born in the United States Among the general population sample 58 stated that they believed President Obama was born in the United States a Find a 90 con dence interval for the di erence between the proportion of Tea Party supporters who believe President Obama was born in the United States and the pro portion …
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