Dec 9 2004 ECON 240A 1 Final L Phillips Answer 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 Margin Smallest 27 3 Q1 40 05 Median 46 Q3 51 875 Largest 62 8 IQR 11 825 a 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 this is 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 Final L Phillips Figure 1 2 Histogram of La Quinta Inn Profit Margins 10 Series PROFITMARGIN Sample 1 100 Observations 100 8 Mean Median Maximum Minimum Std Dev Skewness Kurtosis 6 4 45 73900 46 00000 62 80000 27 30000 7 752131 0 145243 2 349235 2 Jarque Bera Probability 2 116151 0 347123 0 30 35 40 45 50 55 60 d Is this profit margin data significantly different from normal no insignificant Jarque 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 drawn at 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 false negative One test for the presence of aids antibodies is the enzyme linked Dec 9 2004 ECON 240A 3 Final L Phillips immunosorbent 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 the Dec 9 2004 ECON 240A 4 Final L Phillips ELISA test is given a follow up test such as the Western Blot test which is based on a different technology Infected with AIDS Virus ELISA test ELISA test Yes 50 0 50 No 149 9801 9950 199 9801 10 000 Infected with AIDS Virus ELISA test ELISA test Yes P sick P sick P sick 0 005 No P well P well P well 0 995 1 Given P well 0 015 P well P well so P well 0 015 0 995 0 014925 Given P sick 0 003 P sick P sick so P sick 0 003 0 005 0 000015 Infected with AIDS Virus ELISA test ELISA test Yes P sick P sick 0 000015 P sick 0 005 No P well 0 014925 P well P well 0 995 1 The 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 community Dec 9 2004 ECON 240A 5 Final L Phillips 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 seen from their means in Table 3 1 Table 3 1 Table of Means and Other Descriptive statistics The regression results are reported in Table 3 2 Table 3 2 Regression of Profit Margin On Six Explanatory Variables Dependent Variable PROFITMARGIN …
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