Nov. 7, 2006 ECON 240A-1 L. PhillipsMidtermAnswer 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:289 205 244208 218 222255 198 230215 209 222270 341 306222 195206 200221 340210 209224 242209 219250 272222 229213 220220 226250 260209 252247 213273 212217 246a. 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 8 520 8 6 9 9 5 9 0 921 5 0 3 7 8 9 3 222 2 1 4 2 9 0 0 6 2 2 23 024 7 2 6 4Nov. 7, 2006 ECON 240A-2 L. PhillipsMidterm25 5 0 0 226 027 0 3 228 92930 631 323334 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 Moderate HighTempFO-RingFailure Temp FO-RingFailure Temp FO-RingFailure53 yes 68 no 75 yes57 yes 69 no 75 no58 yes 70 no 76 no63 yes 70 yes 76 no66 no 70 yes 78 no67 no 70 no 79 no67 no 72 no 80 no67 no 73 no 81 noa. 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 launchis 0.5, what is the probability of observing exactly one launch with anNov. 7, 2006 ECON 240A-3 L. PhillipsMidtermo-ring failing in the High temperature group? (Hint: you can calculate this or use the Tables in the back of the book). From Table, 0.035Prob(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 conditionalprobability 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 L. PhillipsMidterma. 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 = 0.23b. 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 = 0.17c. 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 = 0.40d. 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 = 0.58e. 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 theTigers.P( WS^WG1) P(WS^LG1) P(WS)P(LS^WG1) P(LS^LG1) P(LS)P(WG1) P9LG1) 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 2006-07. 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 L. PhillipsMidtermThe natural logarithm of California personal income per capita in 2005-06 dollars was regressed against a time index with value zero in 1955-56, one in 1956-57, etc. The results are reported in Table 4-1.Nov. 7, 2006 ECON 240A-6 L. PhillipsMidterma. From the summary statistics reported in Figure 4-1, what type of trend line do you think was added in EXCEL to the figure? _exponential note equation is of form CAPYPC (t) = CAPYPC(0)* ebt where b is theexponential rate of growth__________________b. Why was the natural logarithm of personal income per capita regressedagainst time? _to linearize the exponential growth trend above, take the logarithm of both
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