Dec 11 2001 ECON 240A 1 Final L Phillips Answer all four questions 1 40 points An advertisement in the Singapore Straits Times 2 29 92 featured 48 ladies diamond rings with the weight of the stones in carats and the price in Singapore dollars The data is plotted in Figure 1 A carat is equivalent to 0 2 grams Figure 1 1 Diamond Price 1992 Singapore Vs Weight in Carats 1 Carat 0 2 gm 1200 1000 Price 800 600 400 y 3721x 259 63 2 R 0 9783 200 0 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 Carats a What percent of the total sum of squares for the price of ladies diamond rings is unexplained by the weight of the diamond 2 2 b How many Singapore dollars did the price of a ring go up per carat 3721 Table 1 1 presents some summary statistics and an analysis of variance in tabular form Table 1 1 Regression Summary Statistics and Analysis of Variance SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0 989070664 0 978260778 0 977788186 31 84052227 48 Dec 11 2001 ECON 240A 2 Final L Phillips ANOVA df Regression Residual Total 1 46 47 SS MS 2098595 999 2098596 46635 66747 1013 819 2145231 667 F Significance F 2069 99109 6 75126E 40 c The F statistic is the ratio of what to what explained mean square to unexplained mean square d How many degrees of freedom are in the numerator and how many degrees of freedom are in the denominator 1 and 46 respectively e What is the critical value of the F statistic at the 5 level i e above what value does 5 of the F distribution lie about 4 04 Table 1 2 presents the estimated regression coefficients and their standard errors Table 1 2 Estimated Regression Coefficients with Standard Errors Coefficients Standard Error t Stat Intercept 259 6259072 17 31885619 14 9909 Weight 3721 024852 81 78588037 45 49715 P value 2 5233E 19 6 7513E 40 f The t statistic on weight tests what null hypothesis the coefficient on weight is zero 2 40 points The following data is a random sample of recent owners of Toyota cars One hundred owned Tercels and another hundred owned Avalons The data is crosstabulated with marital status in Table 2 1 Table 2 1 Cross Tabulation of Marital Status with Toyota Model Marital Status Single Married Divorced Widowed Column Sums Tercel Owner 46 37 10 7 100 Avalon Owner 24 52 12 12 100 Row Sums 70 89 22 19 200 Dec 11 2001 ECON 240A 3 Final L Phillips a Under the null hypothesis of independence between marital status and type of Toyota fill in the expected cell counts in Table 2 2 Table 2 2 Expected Cell Counts Under the Hypothesis of Independence Marital Status Single Married Divorced Widowed Column Sums Tercel Owner 35 44 5 11 9 5 Avalon Owner 35 44 5 11 9 5 Row Sums b Using Pearson s Chi Square statistic calculate each cells contribution to Chi Square and enter in Table 2 3 Table 2 3 Each Cell s Contribution to Pearson s Chi Square Marital Status Single Married Divorced Widowed Tercel Owner 3 46 1 26 0 09 0 66 Avalon Owner 3 46 1 26 0 09 0 66 c How many degrees of freedom does Pearson s Chi Square statistic have 3 d Is it significant at the 5 level yes e From a market segmentation and advertising perspective who would you aim your Tercel advertising towards singles Who is your target audience for Avalon advertising married folks 3 40 points On December 30 1986 The Wall Street Journal ran an article about a University of Pittsburgh study investigating whether taller male MBA s earned more than their shorter counterparts The 250 subjects were all about 30 years old and so experience and education are controlled for approximately Of course we would expect income to vary with individual ability which is hard to measure Also income could vary with gender if there was discrimination but this is controlled for as well by restricting the sample to males The MBA males were polled to report their height in inches and their annual incomes The results are plotted in Figure 3 1 with a fitted linear regression line illustrated Dec 11 2001 ECON 240A 4 Final L Phillips Figure 3 1 Annual Salary Versus Height in Inches for 250 Male MBA s About 30 Years Old 90000 80000 70000 Annual Salary in 60000 50000 40000 30000 y 604 11x 17933 2 R 0 0505 20000 10000 0 0 10 20 30 40 50 60 70 80 Height in Inches a Do you have enough information from the chart to test whether annual salary is affected by height yes b If so what is the distribution for the test statistic F c Is it significant at the 5 level yes d How about the 1 level yes e What is your interpretation of the slope coefficient 604 of annual salary per inch of height 90 Dec 11 2001 ECON 240A 5 Final L Phillips 4 40 points One way to estimate one way single factor analysis of variance ANOVA is with a tabular approach dividing the total sum of squares into the explained sum of squares plus the unexplained sum of squares An alternative but equivalent methodology is regression If the single factor has two classes this can be captured by an indicator dummy variable which takes on value one if the observation is in the first class and value zero if the observation is in the second class We could use ANOVA in regression form to see if o ring failure was temperature dependent The dummy explanatory variable Bern has two classes the first class for launches with one or more o ring failures and the second class for zero o ring failures with Bern taking on values of one or zero respectively In the regression of launch temperature against a constant term and Bern as shown in the table of results Variable Coefficient Std Error t Statistic Prob C BERN 72 58824 8 873950 1 471245 2 724217 49 33796 3 257432 0 0000 0 0036 0 325378 0 294714 6 066099 809 5462 76 27557 0 575304 Mean S D Akaike Schwarz F statistic Prob F R squared Adjusted R squared S E of regression Sum squared resid Log likelihood Durbin Watson stat 70 00000 7 223151 6 522964 6 621135 10 61086 0 003608 a How do you interpret the constant term mean temperature for successful launches b How do you interpret the coefficient on Bern difference in mean temperature for launches with o ring failures and mean temperature for successful launches c What statistic would you use to test the null hypothesis that the mean temperature is the same for the two classes t statistic on Bern d What is the unexplained sum of squares for this single factor ANOVA analysis 809 55
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