Nov 7 2006 ECON 240A 1 Midterm L Phillips Answer all four questions 1 20 A list of housing prices in thousands for homes sold in a recent year for a suburb in Los Angeles follows 289 208 255 215 270 222 206 221 210 224 209 250 222 213 220 250 209 247 273 217 205 218 198 209 341 195 200 340 209 242 219 272 229 220 226 260 252 213 212 246 244 222 230 222 306 a How many observations are there 45 b What is the minimum price 195 What is the maximum price 341 c Draw a stem and leaf diagram To save time you do not have to order the leaves d Define and provide a numerical value of the central tendency median 222 e Is this distribution symmetric no skewed to high prices 19 8 5 20 8 6 9 9 5 9 0 9 21 5 0 3 7 8 9 3 2 22 2 1 4 2 9 0 0 6 2 2 23 0 24 7 2 6 4 Nov 7 2006 ECON 240A 2 Midterm L Phillips 25 5 0 0 2 26 0 27 0 3 2 28 9 29 30 6 31 32 33 34 1 0 2 20 The data for shuttle launches prior to the Challenger disaster follows The data is divided into three groups by launch temperature Low 53 67 degrees Fahrenheit Moderate 68 73 degrees Fahrenheit and High 75 81 degrees Fahrenheit As discussed in class a no means there were zero o ring failures for the launch A yes means there were one or more o ring failures Low Temp F 53 57 58 63 66 67 67 67 Moderate O Ring Failure yes yes yes yes no no no no Temp F 68 69 70 70 70 70 72 73 High O Ring Failure no no no yes yes no no no Temp F 75 75 76 76 78 79 80 81 O Ring Failure yes no no no no no no no a What is the probability of a launch with an o ring failing for the Low temperature sub sample 0 5 b What is the probability of a launch with an o ring failing for the Moderate temperature sub sample 0 25 c What is the probability of a launch with an o ring failing for the High temperature sub sample 0 125 d Assume the probability that one or more o rings fails in a given launch is 0 5 what is the probability of observing exactly one launch with an Nov 7 2006 ECON 240A 3 Midterm L Phillips o ring failing in the High temperature group Hint you can calculate this or use the Tables in the back of the book From Table 0 035 Prob k 1 8 7 1 0 57 0 5 1 32 0 031 Note the book is actually inaccurate i e a misprint but close enough for government work e Assuming that the probability that one or more o rings fails in a given launch is 0 5 like flipping a coin do you think that observing the event of one or fewer launches with an o ring failure for the High temperature group is an unusual event What is your criterion for unusual Yes The prob k 0 0 004 so Prob k 2 035 which is less than 5 the norm for a Type I error f In part e the null hypothesis is that the probability p of observing a launch in the High temperature group with one or more o ring failures is 0 5 The alternative hypothesis is that this probability p 0 5 You can either accept this null or reject it Like President Bush you are the decision maker What are the implications or costs of accepting this null if it is false Discuss Accepting the null is equivalent to accepting that launches with o ring failures were not temperature dependent This would support proceeding with the launch of the Challenger in freezing temperatures twenty degrees below any previous launch This type II error accepting the null when it is false has a high cost in lives money time and public confidence in NASA 3 20 Before the first game of this Fall s World Series between the Detroit Tigers and the St Louis Cardinals the Tiger s were two to one favorites to win the series i e the probability the Cardinals would win the series was 0 33 From the historical record suppose the conditional probability of winning the first game given the team ends up winning the series is 0 70 while the conditional probability of winning the first game given the team ends up losing the series is 0 25 This Fall the Cardinals won the first game in Detroit Nov 7 2006 ECON 240A 4 Midterm L Phillips a From the information above what was the joint probability of the Cardinals winning the first game and then going on to win the World Series round to two decimal points p WG1 WS P WG1 WS P WS 0 70 0 33 0 23 b From the information above what was the joint probability of the Cardinals winning the first game and then going on to lose the World Series round to two decimal points p WG1 LS P WG1 LS P LS 0 25 0 67 0 17 c What is the marginal probability of the Cardinals winning the first game i e whether they end up winning or losing the World Series P WG1 P WG1 WS P WG1 LS 0 23 0 17 0 40 d What was the conditional probability of the Cardinals going on to win the World Series given they won the first game P WS WG1 P WS WG1 P WG1 0 23 0 40 0 58 e If you had bet 100 on the Tigers to win 50 before the first game what should you do after the Cardinals win the first game Bet on the Cardinals Now the odds are approximately 60 to 40 so you will have to bet 150 to win 100 which will then pay off your losing bet on the Tigers P WS WG1 P LS WG1 P WG1 P WS LG1 P LS LG1 P9LG1 P WS P LS 1 Hint fill in the two by two boxes of joint probabilities bordered by the marginal probabilities on the right and bottom 4 20 The California Department of Finance web site has the California Consumer Price Index for urban consumers CPI U from fiscal year 1955 56 through 200607 Data for personal income and population in the state can be obtained from Schedule 6 of the Governor s Budget Summary also available at the same site for fiscal years These data were used to calculate California personal income per capita in 2005 06 constant dollars A plot of this variable against fiscal year is shown in Figure 4 1 Nov 7 2006 ECON 240A 5 Midterm L Phillips The natural logarithm of California personal income per capita in 2005 06 dollars was regressed …
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