1 An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are: Probability of at least one Type of Driver % of all Drivers collision Teenager 8 0.15 Young Adult 16 0.08 Middle Age 45 0.04 Senior 31 0.05 a) What is the probability that a driver is involved in a collision given that they are middle aged? b) What is the probability that a driver is involved in at least one collision? c) A person phones the insurance company to make a claim and therefore we can assume that they have been involved in a collision. What is the probability that they are a young adult? Problem 2 Choi, Laibson and Madrian conducted an experiment to study the choices made in fund selection. Suppose 100 undergraduate students and 100 MBA students were selected. When presented with 4 S&P 500 Index funds that were identical expect for their fees, undergraduate and MBA students chose the funds as follows: Student Group FUND Undergraduates MBA Lowest cost fund 19 19 Second lowest cost fund 37 40 Third lowest cost fund 17 23 Highest cost fund 27 18 www.euruni.edu 1 If a student is selected at random, what is the probability that he or she a) b) c) d) selected the lowest or second lowest cost fund? Selected the lowest cost fund and is an undergraduate? Selected the lowest cost fund or is an undergraduate? Given that a student is an undergraduate, what is the probability that he or she selected the highest cost fund? Problem 3. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005). a) In a sample of 10 workers, what is the probability that exactly three workers take public transportation daily? b) In a sample of 10 workers, what is the probability that at least three workers take public transportation daily? Problem 4. Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here (Barron's, January 23, 2006; February 13, 2006; and February 27, 2006). 214 163 265 194 180 202 198 212 201 174 171 211 211 The probability distribution of trading volume is approximately normal. www.euruni.edu 2 a) Compute the mean and standard deviation to use as estimates of the population mean and standard deviation. b) What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares? c) What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares? Problem 5 The Dow Jones Travel Index reported what business travelers pay for hotel rooms per night in major U.S. cities (The Wall Street Journal, January 16, 2004). The average hotel room rates for 20 cities are as follows: Atlanta $163 Minneapolis $125 Boston 177 New Orleans 167 Chicago 166 New York 245 Cleveland 126 Orlando 146 Dallas 123 Phoenix 139 Denver 120 Pittsburgh 134 Detroit 144 San Francisco 167 Houston 173 Seattle 162 Los Angeles 160 St. Louis 145 Miami 192 Washington, D.C. 207 a) What is the mean hotel room rate? b) What is the median hotel room rate? c) What is the mode? www.euruni.edu 3 d) What is the first quartile? What is the meaning of first quartile? e) What is the third quartile? What is the meaning of first quartile? www.euruni.edu 4 Problem 1 An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are: Probability of at least one Type of Driver % of all Drivers collision Teenager 8 0.15 Young Adult 16 0.08 Middle Age 45 0.04 Senior 31 0.05 a) What is the probability that a driver is involved in a collision given that they are middle aged? b) What is the probability that a driver is involved in at least one collision? c) A person phones the insurance company to make a claim and therefore we can assume that they have been involved in a collision. What is the probability that they are a young adult? Problem 2 Choi, Laibson and Madrian conducted an experiment to study the choices made in fund selection. Suppose 100 undergraduate students and 100 MBA students were selected. When presented with 4 S&P 500 Index funds that were identical expect for their fees, undergraduate and MBA students chose the funds as follows: Student Group FUND Undergraduates MBA Lowest cost fund 19 19 Second lowest cost fund 37 40 Third lowest cost fund 17 23 Highest cost fund 27 18 www.euruni.edu 1 If a student is selected at random, what is the probability that he or she a) b) c) d) selected the lowest or second lowest cost fund? Selected the lowest cost fund and is an undergraduate? Selected the lowest cost fund or is an undergraduate? Given that a student is an undergraduate, what is the probability that he or she selected the highest cost fund? Problem 3. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005). a) In a sample of 10 workers, what is the probability that exactly three workers take public transportation daily? b) In a sample of 10 workers, what is the probability that at least three workers take public transportation daily? Problem 4. Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here (Barron's, January 23, 2006; February 13, 2006; and February 27, 2006). 214 163 265 194 180 202 198 212 201 174 171 211 211 The probability distribution of trading volume is approximately normal. www.euruni.edu 2 a) Compute the mean and standard deviation to use as estimates of the population mean and standard deviation. b) What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares? c) What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares? Problem 5 The Dow Jones Travel Index reported what business travelers pay for hotel rooms per night in major U.S. cities (The Wall Street Journal, January 16, 2004). The average hotel room rates for 20 cities are as follows: Atlanta $163 Minneapolis $125 Boston 177 New Orleans 167 Chicago 166 New York 245 Cleveland 126 Orlando 146 Dallas 123 Phoenix 139 Denver 120 Pittsburgh 134 Detroit 144 San Francisco 167 Houston 173 Seattle 162 Los Angeles 160 St. Louis 145 Miami 192 Washington, D.C. 207 a) What is the mean hotel room rate? b) What is the median hotel room rate? c) What is the mode? www.euruni.edu 3 d) What is the first quartile? What is the meaning of first quartile? e) What is the third quartile? What is the meaning of first quartile? www.euruni.edu 4 \fProblem 1 An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are: Probability of at least one Type of Driver % of all Drivers collision Teenager 8 0.15 Young Adult 16 0.08 Middle Age 45 0.04 Senior 31 0.05 a) What is the probability that a driver is involved in a collision given that they are middle aged? b) What is the probability that a driver is involved in at least one collision? c) A person phones the insurance company to make a claim and therefore we can assume that they have been involved in a collision. What is the probability that they are a young adult? Problem 2 Choi, Laibson and Madrian conducted an experiment to study the choices made in fund selection. Suppose 100 undergraduate students and 100 MBA students were selected. When presented with 4 S&P 500 Index funds that were identical expect for their fees, undergraduate and MBA students chose the funds as follows: Student Group FUND Undergraduates MBA Lowest cost fund 19 19 Second lowest cost fund 37 40 Third lowest cost fund 17 23 Highest cost fund 27 18 www.euruni.edu 1 If a student is selected at random, what is the probability that he or she a) b) c) d) selected the lowest or second lowest cost fund? Selected the lowest cost fund and is an undergraduate? Selected the lowest cost fund or is an undergraduate? Given that a student is an undergraduate, what is the probability that he or she selected the highest cost fund? Problem 3. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005). a) In a sample of 10 workers, what is the probability that exactly three workers take public transportation daily? b) In a sample of 10 workers, what is the probability that at least three workers take public transportation daily? Problem 4. Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here (Barron's, January 23, 2006; February 13, 2006; and February 27, 2006). 214 163 265 194 180 202 198 212 201 174 171 211 211 The probability distribution of trading volume is approximately normal. www.euruni.edu 2 a) Compute the mean and standard deviation to use as estimates of the population mean and standard deviation. b) What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares? c) What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares? Problem 5 The Dow Jones Travel Index reported what business travelers pay for hotel rooms per night in major U.S. cities (The Wall Street Journal, January 16, 2004). The average hotel room rates for 20 cities are as follows: Atlanta $163 Minneapolis $125 Boston 177 New Orleans 167 Chicago 166 New York 245 Cleveland 126 Orlando 146 Dallas 123 Phoenix 139 Denver 120 Pittsburgh 134 Detroit 144 San Francisco 167 Houston 173 Seattle 162 Los Angeles 160 St. Louis 145 Miami 192 Washington, D.C. 207 a) What is the mean hotel room rate? b) What is the median hotel room rate? c) What is the mode? www.euruni.edu 3 d) What is the first quartile? What is the meaning of first quartile? e) What is the third quartile? What is the meaning of first quartile? www.euruni.edu 4 \fProblem 1 Probability of at least one collision 8% 0.15 16% 0.08 45% 0.04 31% 0.05 Type of Driver Teenager Young Adult Middle Age Senior % of all drivers a) P(collision| middle aged) 0.04 b) P(a driver is involved in at least one collision) 0.0583 c) P(young adult| collision) 0.2196 Problem 2 Student Group FUND Undergraduates MBA Lowest cost fund 19 19 Second lowest cost fund 37 40 Third lowest cost fund 17 23 Highest cost fund Total 27 100 18 100 a) P(lowest or second lowest cost fund) 0.575 b) P(lowest cost fund and an undergraduate) 0.095 c) P(lowest cost fund or an undergraduate) 0.595 d) P(highest cost fund| an undergraduate) 0.27 Problem 3 n 10 200 p q 30% 70% a) P( 3 workers) 0.266827932 b) P( 0 workers) 0.0282475249 P( 1 workers) 0.121060821 P( 2 workers) 0.2334744405 P(at least three workers) 0.6172172136 Problem 4 214 202 174 a) 163 198 171 Mean 199.6923076923 Standard deviation 26.0396640256 b) P(X<180) 0.2247518265 c) P(X>230) 0.1222312787 Problem 5 Atlanta Boston Chicago Cleveland Dallas Denver Detroit Houston Losangeles Miami Minneapolis New Orleans New York Orlando Phoenix Pittsburgh San Francisco 265 212 211 163 177 166 126 123 120 144 173 160 192 125 167 245 146 139 134 167 194 201 211 180 Seattle St. Louis Washington DC a) b) c) d) e) 162 145 207 Mean Median Mode First quartile Third quartile 159.05 161 167 137.75 168.5 There are 25% of data lower than the first quartile. There are 75% of data lower than the first quartile. ower than the first quartile. ower than the first quartile