Nov 2 2000 ECON 240A 1 Midterm L Phillips Answer all five questions 1 15 points Approximately three out of four Americans who filed a 1995 tax return received a refund If three individuals are cchosen at random from among those who filed a 1995 tax return find the probabilities of the following events a All three received a refund b None of the three received a refund c Exactly one received a refund P 3 4 1 p 1 4 refund No refund Using the binomial a prob k 3 n k n k pk 1 p n k 3 3 0 3 4 3 1 4 0 27 64 b prob k 0 3 0 3 3 4 0 1 4 3 1 64 c Prob k 1 3 1 2 3 4 1 1 4 2 9 64 2 15 points For the test of the hypothesis H0 1000 Ha 1000 Given 0 05 200 n 100 a Find when 900 The z statistic is z x n x 1000 200 10 1 96 and solving for x we obtain 960 8 Thus the decision rule from the null hypothesis is accept the null if x 960 8 and reject the null if x 960 8 Note for z 1 96 solving for x we obtain 1039 2 so far above a population mean of 900 that it will not affect the calculation of If the mean of the population were 900 then using this decision rule the probability of accepting the null when false i e the probability of the type II error is Nov 2 2000 ECON 240A 2 Midterm L Phillips the area above 960 8 when the mean is 900 i e z 960 8 900 20 3 04 so from the standardized normal distribution the area above z 3 04 is 0 0012 3 15 points A certain city has one morning newspaper and one evening newspaper It is estimated that 20 of the city s households subscribe to the morning newspaper and 60 subscribe to the evening paper Of those who subscribe to the morning paper 80 also subscribe to the evening paper a What proportion of households subscribe to both papers b What proportion of households subscribes to at most one of the papers c What proportion of households subscribes to neither paper P M E 0 16 Morning Prob M 0 2 Evening Prob E 0 6 M 0 2 E 0 6 Evening Given Morning Prob E M 0 8 a P E M P E M P M 0 8 0 2 0 16 b Probability you subscribe to at most one is one minus the probability you subscribe to both i e 1 P E M 1 0 16 0 84 c The probability you subscribe to neither is one minus the probability you subscribe to either one or the other i e 1 P M E where P M E P M P U P M I E 0 2 0 6 0 16 0 64 so the probability of subscribing to neither is 0 36 It is the area outside the circles in the Venn diagram accounting for the area of overlap Nov 2 2000 ECON 240A 3 Midterm L Phillips 4 15 points The following table shows the results of regressing the natural logarithm of California General Fund expenditures in billions of nominal dollars against year beginning in 1968 and ending in 2000 A plot of actual estimated and residual values follows a How much of the variance in the dependent variable is explained by trend b What is the meaning of the F statistic in the table Is it significant c Interpret the estimated slope d If General Fund expenditures was 68 819 billion in California for fiscal year 2000 2001 provide a point estimate for state expenditures for 2001 2002 e A state senator believes that state expenditures in nominal dollars have grown over time at 7 a year Is the senator in the ballpark or is his impression significantly below the estimated rate using a 5 level of significance f If you were an aid to the Senator how might you criticize this regression Table Dependent Variable LNGENFND Method Least Squares Sample 1968 2000 Included observations 33 Variable Coefficient Std Error t Statistic Prob YEAR C 0 086958 169 4787 0 003895 7 726922 22 32804 21 93353 0 0000 0 0000 R squared Adjusted R squared S E of regression Sum squared resid Log likelihood Durbin Watson stat 0 941459 0 939570 0 213030 1 406835 5 235258 0 118575 Mean dependent var S D dependent var Akaike info criterion Schwarz criterion F statistic Prob F statistic 3 046404 0 866594 0 196076 0 105379 498 5416 0 000000 a From R2 94 1 b The F statistic is the ratio of the explained mean square to the unexplained mean square and so is a test of the explanatory power of the regression The F statistic is very large and highly significant Note F ESS k 1 USS n k R2 TSS k 1 1 R2 TSS n k R2 1 R2 n k k 1 Nov 2 2000 c d e f ECON 240A 4 Midterm L Phillips so the F statistic and the coefficient of determination are conveying the same information The slope 0 087 is the annual rate of growth of General Fund expenditures or in percent per annum 8 7 The point estimate is 68 819 B 1 087 74 806 B The t statistic is t 0 087 0 07 0 0039 4 36 and is highly significant so the Senator s hypothesis is rejected As the Senator s assistant you could point out that the assumptions for ordinary least squares are not satisfied since from the plot the residuals are positively correlated reflected as well by the Durbin Watson statistic of 0 12 way below 2 Plot Actual Fitted and Residual Values from the Regression of the Logarithm of General Fund Expenditures B on Year 5 4 3 0 4 2 0 2 1 0 0 0 2 0 4 70 75 80 Residual 85 Actual 90 95 00 Fitted 5 15 points The monthly rate of return for the Gillette stock is regressed against the monthly rate of return for the Standard and Poor s Composite Index and the results are reported in the table below A plot of actual estimated and residual values follows Nov 2 2000 ECON 240A 5 Midterm L Phillips a Interpret the economic meaning of the estimated coefficient on the Standard and Poor s Composite Index b Is this estimated coefficient significantly different from zero at the 5 level Explain c Is this estimated coefficient significantly different from one at the 5 level Explain d What is the economic significance of the test in part b What is the economic significance of the test in part c e Interpret the economic meaning of the estimated coefficient of determination f What do you conclude from the combination of the information in R2 and the information in the plot of actual fitted and residual values Table Dependent Variable GILLETTE …
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