Marital StatusTercel OwnerAvalon OwnerRow SumsColumn SumsTercel OwnerAvalon OwnerRow SumsColumn SumsMarital StatusTercel OwnerAvalon OwnerDec. 11, 2001 ECON 240A-1 L. PhillipsFinalAnswer 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.--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------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 OUTPUTRegression StatisticsMultiple R 0.989070664R Square 0.978260778Adjusted R Square 0.977788186Standard Error 31.84052227Observations 48Figure 1.1 Diamond Price (1992 Singapore $) Vs. Weight in Carats (1 Carat = 0.2 gm)y = 3721x - 259.63R2 = 0.97830200400600800100012000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4CaratsPriceDec. 11, 2001 ECON 240A-2 L. PhillipsFinalANOVAdf SS MS F Significance FRegression 1 2098595.999 2098596 2069.99109 6.75126E-40Residual 46 46635.66747 1013.819Total 47 2145231.667c. 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 P-valueIntercept -259.6259072 17.31885619 -14.9909 2.5233E-19Weight 3721.024852 81.78588037 45.49715 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 cross-tabulated with marital status in Table 2.1.-------------------------------------------------------------------------------------------------------Table 2.1: Cross-Tabulation of Marital Status with Toyota ModelMarital Status Tercel Owner Avalon Owner Row SumsSingle 46 24 70Married 37 52 89Divorced 10 12 22Widowed 7 12 19Column Sums 100 100 200------------------------------------------------------------------------------------------------------------Dec. 11, 2001 ECON 240A-3 L. PhillipsFinala. 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 IndependenceMarital Status Tercel Owner Avalon Owner Row SumsSingle 35 35Married 44.5 44.5Divorced 11 11Widowed 9.5 9.5Column Sumsb. 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-SquareMarital Status Tercel Owner Avalon OwnerSingle 3.46 3.46Married 1.26 1.26Divorced 0.09 0.09Widowed 0.66 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 L. PhillipsFinal -------------------------------------------------------------------------------------------------------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________________Figure 3.1: Annual Salary Versus Height in Inches for 250 Male MBA's About 30 Years Oldy = 604.11x + 17933R2 = 0.050501000020000300004000050000600007000080000900000 10 20 30 40 50 60 70 80 90Height in InchesAnnual Salary in $Dec. 11, 2001 ECON 240A-5 L. PhillipsFinal 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
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