Dec 11 2008 Final ECON 240A 1 L Phillips Answer all five questions 1 30 Using Titanic3 the data set for Project 1 809 people perished and 500 people survived for a total of 1309 Figure 1 1 is a plot of the histogram for fare in pounds sterling for the people who perished and Figure 1 2 is a plot of the histogram for fare for the people who survived Figure 1 1 Histogram of Fare in Pounds Sterling for People Who Perished 400 Series FARE Sample 1 809 Observations 808 300 200 100 Mean Median Maximum Minimum Std Dev Skewness Kurtosis 23 35383 10 50000 263 0000 0 000000 34 14510 4 248942 24 56307 Jarque Bera Probability 18085 07 0 000000 0 0 40 80 120 160 200 240 Figure 1 2 Histogram of Fare in Pounds Sterling of People who Survived 250 Series FARE Sample 810 1309 Observations 500 200 150 100 Mean Median Maximum Minimum Std Dev Skewness Kurtosis 49 36118 26 00000 512 3292 0 000000 68 64880 3 518007 19 73316 Jarque Bera Probability 6864 669 0 000000 50 0 0 50 100 150 200 250 300 350 400 450 500 a Looking at two measures of central tendency does fare appear to distinguish the two groups those who survived and those who perished The mean and median fare for those who survived are both more than twice the amount of the mean and median fare for those who perished Dec 11 2008 Final ECON 240A 2 L Phillips b The statistics accompanying these two figures provide two separate measures of dispersion Name these two measures and explain which you think is the most reliable and why that is the case Standard deviation and range The standard deviation is likely more reliable since the range i e the max min can be influenced by outliers especially the max in this case c A measure of relative dispersion different from the two quantitative measures referred to in part b is the sample coefficient of variation Calculate and report this measure for those who perished 1 46 and for those who survived 1 39 Does there appear to be much of a difference in relative dispersion between survivor fare and perished fare Yes or no no d For which group is the fare distribution most skewed perished or survived perished What is used as a measure of skewness the first central moment Explain The distributions of fare for both those who survived and those who did not are quite nonnormal There were 127 women who perished Their distribution for fare in pounds sterling is displayed in the Box plot Figure 1 3 Figure 1 3 Box Plot of Fare in for 127 Women who Perished on the Titanic e Comment on a salient feature of the distribution displayed in figure 1 3 outliers 2 30 After looking at the data in question 1 a statistician has an inspired idea Why not regress fare for both those who survived and those that perished against a constant and a dummy or indicator variable that is one if the individual perished and zero otherwise and see what happens The results are shown in Table 2 1 Table 2 1 Regression of Fare Against an Indicator Variable of Perished Or Not Dependent Variable FARE Method Least Squares Sample 1 1309 Included observations 1308 Excluded observations 1 Variable Coefficient Std Error t Statistic Prob Dec 11 2008 Final PERISHED C R squared Adjusted R squared S E of regression Sum squared resid Log likelihood Durbin Watson stat ECON 240A 3 26 00735 49 36118 0 059666 0 058946 50 21003 3292487 6977 380 1 836247 2 856957 2 245461 L Phillips 9 103167 21 98265 Mean dependent var S D dependent var Akaike info criterion Schwarz criterion F statistic Prob F statistic 0 0000 0 0000 33 29548 51 75867 10 67183 10 67975 82 86765 0 000000 The statistician notes the regression appears to be highly significant but is somewhat surprised to see the coefficient on the indicator variable is negative since the dependent variable fare is always non negative In any case the statistician is not quite sure what this regression means and comes to you with a list of questions a What is the meaning of this regression and the fact that the F statistic is significant i e what is the null hypothesis that this regression is testing The significance of the regression is that the average fare for those who survived is significantly different from the mean fare of those who perished so reject the null of no difference b How should the statistician interpret the constant term The constant is the mean fare for those who survived see Figure 1 2 Fare c b perished e E Fare perished 0 c 0 0 where left hand side is mean fare for survivors c Explain to the statistician why the coefficient on the indicator variable is negative and what this coefficient measures E fare perished 1 c b 0 so c mean fare for perished mean fare for survivors and so b 0 see Fig 1 1 and Fig 1 2 d 1If the statistician questioned your explanation how else could you demonstrate the validity of your answers You could regress fare against two indicator variables with no constant fare b survivor d perished e and show that b was the mean fare for survivors and d the mean fare for those who perished and run a Wald test on b d and show that the F stat is the same as for the regression You could test the difference between means between two populations e You need not do the calculations but show the formula you would use from Ch 13 supposing you chose that path to answer part d t x1 x2 u1 u2 s12 n1 s22 n2 1 2 3 30 The Rapid Test also known as ELISA is used to determine whether someone has HIV the virus that causes AIDS Information from the Centers for Disease Control CDC can be found at www cdc gov hiv resources qa oraqck htm The conditional probability of getting a positive test result for someone who does not have the virus i e a false positive is 0 027 The conditional probability of getting a negative test result for someone who does have the virus i e a false negative is 0 080 On the basis of several characteristics a doctor has assigned a patient as being low risk i e as having a probability of having the virus of 0 005 Then the doctor and patient receive the Rapid Test result for this patient as being positive Dec 11 2008 Final ECON 240A 4 L Phillips a What the patient and his doctor both want to know is what is the probability that this patient actually has the HIV virus report to the third decimal place P HIV P HIV P b Before answering part a it might be easier to use the false positive rate to calculate the joint probability the patient does not have the HIV virus and tests positive Calculate to …
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