Dec. 9, 2004 ECON 240A-1 L. PhillipsFinalAnswer all five questions.1. (30 minutes) The operating margin for a sample of 100 La Quinta Inns is displayed in a Box plot in Figure 1-1. The operating margin is the sum of profits, depreciation and interest expenses as a percent of total revenue. Some of the Box plot statistics are reported in Table 1-1.Figure 1-1: Boxplot for the Operating Margin for 100 La Quinta Inns. Table 1-1: Boxplot statistics for Figure 1-1.MarginSmallest = 27.3Q1 = 40.05Median = 46Q3 = 51.875Largest = 62.8IQR = 11.825a. What is the central tendency for this data? __median of 46 _______b. What is the measure of dispersion for this data? __Interquartile range of 11.8 c. Explain the location of the end of the upper whisker_Q3 + 1.5*IQR, since thisis more than the largest observation of 62.8, the whisker ends at 62.8 The histogram of the profit margin data is illustrated in Figure 1-2.Dec. 9, 2004 ECON 240A-2 L. PhillipsFinal024681030 35 40 45 50 55 60Series: PROFITMARGINSample 1 100Observations 100Mean 45.73900Median 46.00000Maximum 62.80000Minimum 27.30000Std. Dev. 7.752131Skewness -0.145243Kurtosis 2.349235Jarque-Bera 2.116151Probability 0.347123Figure 1-2: Histogram of La Quinta Inn Profit Marginsd. Is this profit margin data significantly different from normal? _no, insignificantJarque-Bera stat ___________ e. If profit margin was regressed against some explanatory variables, what distribution would you expect for the error term? Explain. _normal, likely the same as the dependent variable ________________ ________________f. Is the measure of dispersion reported in Fig. 1-2 larger or smaller than the measure of dispersion reported in Fig. 1-1? _Std. Dev. of 7.8 is smaller than IQR____________________________2. (30) On the prevalence of Aids, the Centers for Disease Control reference an article, “ HIV in the United States At the Turn of the Century: An Epidemic in Transition”, American Journal of Public Health, July 2001. The authors, John Keron et. al., estimate between 800,000 and 900,000 people were infected with the aids virus at the turn of the century. Of course, prevalence varies with gender, race, region, behavior, and socioeconomic status. We will consider a person drawnat random from the general population at risk, i.e. those in the nation approximately 14 years of age and older, and use the probability of being infected as 0.005 or 1 in 200, based on the authors’ estimate and the population by age estimates from the U.S. Census. Two types of errors can be made in diagnostic tests for a disease. A well person may test positive (false positive) and a sick person may test negative (falsenegative). One test for the presence of aids antibodies is the enzyme-linkedDec. 9, 2004 ECON 240A-3 L. PhillipsFinalimmunosorbent assay (ELISA) test. A person may take up to six months to generate anti-bodies after being infected, so timing may affect results. SANOCHEMIA corporation has a web site, http://www.flourognost.com/REFERENCES/AIDS%20testing_information.htm that provides some characteristics for the ELISA test. “Specificity” of the test is the conditional probability of a false positive and it is 0.015. “Sensitivity” of the test is the conditional probability of a false negative and it is 0.003.a. Given that a person’s ELISA test is positive, what is the probability the person is well? _P(Well/+) = 0.75 =P(well+)/P(+), where given P(+/well)= 0.015 = P(well+)/P(well) where given P(well) = 0.995, so P(well+) = 0.014925. P(+) = P(well+) + P(sick+). P(sick+) =P(sick) –P(sick-) =0.005-–P(sick-) . P(-/sick) is given as 0.003 = P(sick-)/P(sick) where P(sick) is 0.005 so P(sick-) = 0.003*0.005 = 0.000015. Or you can use the tabular approach below.______________________b. Given a person’s ELISA test is positive, what is the probability the person is sick? _P(sick/+) = 0.25 since given a + test you are either wellor sick. And it can be calculated analogous to part a__________________c. If 10,000 persons were drawn from the general national population at random, how many would fall in each black bordered box below? Round to the nearest integer.d. Based on your answers to parts a and b above, and the numbers in the boxes (row) for ELISA test +, explain the false positive paradox. __Most people are well so even though the test has pretty good specificity and sensitivity most people will be well and test negative and the number that are well and test positive will exceed the number that are sick and test negative. Note only 50 of the 10,000 are sick and all of thes test positive, none test negative.Note: Because of this false positive paradox, in spite of the accuracy of the ELISA test, as measured by its specificity and sensitivity, and because of the serious implications of harboring the virus, a person testing positive on theDec. 9, 2004 ECON 240A-4 L. PhillipsFinalELISA test is given a follow-up test, such as the Western Blot test, which is based on a different technology.Infected with AIDS VirusYes No ELISA test + 50 149 199 ELISA test - 0 9801 980150 9950 10,000Infected with AIDS VirusYes No ELISA test + P(sick+) P(well+) ELISA test - P(sick-) P(well-)P(sick) = 0.005 P(well) = 0.995 1Given P(+/well) = 0.015 = P(well+)/P(well), so P(well+) = 0.015*0.995 =0.014925Given P(-/sick) = 0.003 = P(sick-)/P(sick), so P(sick-) = 0.003*0.005 =0.000015Infected with AIDS VirusYes No ELISA test + P(sick+) P(well+)=0.014925 ELISA test - P(sick-)=0.000015 P(well-)P(sick) = 0.005 P(well) = 0.995 1The rest can be filled in by subtraction and addition, and these probabilities multiplied by 10,000 and rounded to give the numbers above.3. (30) The profit margin data for La Quinta Inns displayed in problem 1, was regressed against a number of explanatory variables including:(1) Number: the number of competitive motel and hotel rooms within 3 miles of the inn.(2) Nearest: the number of miles to the closest competitor(3) Officespace: A demand variable of the thousands of square feet of office space in the surrounding communityDec. 9, 2004 ECON 240A-5 L. PhillipsFinal(4) Enrollment: college and university enrollment in thousands in nearby schools(5) Income: median household income in thousands in the surrounding community(6) Distance: from the inn to downtown in miles The scale and units of the variables can be
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