Answer all five questionsTablea. From R2, 94.1 %.PlotTablePlotNov. 2, 2000 ECON 240A-1 L. PhillipsMidtermAnswer all five questions1. (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 refundb. None of the three received a refund.c. Exactly one received a 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 = 1/64c. Prob(k=1) = 3!/1!2! (3/4)1(1/4)2 = 9/642. (15 points) For the test of the hypothesis,H0 : = 1000Ha : 1000Given = 0.05, = 200, n = 100a. Find when = 900The z statistic is: z = (x - )/(/n) = (x -1000)/(200/10) = -1.96, and solving forx , we obtain 960.8. Thus the decision rule from the null hypothesis is, accept the null ifx >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 refundNo refundP =3/41 – p = 1/4Nov. 2, 2000 ECON 240A-2 L. PhillipsMidtermthe 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?------------------------------------------------------------------------------------------------a. P(EM) = P(E/M)*P(M) = 0.8*0.2 = 0.16b. Probability you subscribe to at most one is one minus the probability you subscribe to both, i.e. = 1 – P(EM) = 1 – 0.16 = 0.84c. 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 IE) = 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.Morning, Prob(M) = 0.2Evening, Prob(E) = 0.6Evening Given Morning, Prob (E/M) = 0.8M0.2E0.6P(ME) = 0.16Nov. 2, 2000 ECON 240A-3 L. PhillipsMidterm4. (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? TableVariable Coefficient Std. Error t-Statistic Prob. YEAR 0.086958 0.003895 22.32804 0.0000C -169.4787 7.726922 -21.93353 0.0000R-squared 0.941459 Mean 3.046404Adjusted R-squared 0.939570 S.D. 0.866594S.E. of regression 0.213030 Akaike -0.196076Sum squared resid 1.406835 Schwarz -0.105379Log likelihood 5.235258 F-statistic 498.5416Durbin-Watson stat 0.118575 Prob(F- 0.000000a. 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 Fstatistic 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)so the F statistic and the coefficient of determination are conveying the same information.c. The slope, 0.087, is the annual rate of growth of General Fund expenditures, or in percent per annum, 8.7%d. The point estimate is $68.819 B * 1.087 = $74.806 B.Nov. 2, 2000 ECON 240A-4 L. PhillipsMidterme. The t statistic is t = (0.087 – 0.07)/0.0039 = 4.36 and is highly significant so the Senator’s hypothesis is rejected.f. As the Senator’s assistant, you could point out that the assumptions for ordinary least squares are not satisfied since from the plot the residuals are positively correlated, reflected as well by the Durbin-Watson statistic of 0.12, way below 2.Plot5. (15 points) The monthly rate of return for the Gillette stock is regressed against the monthly rate of return for the Standard and Poor’s Composite Index, and the results are reported in the table below. A plot of actual, estimated and residual values follows.a. Interpret the economic meaning of the estimated coefficient on the Standard and Poor’s Composite Index.b. Is this estimated coefficient significantly different from zero, at the 5% level? Explain.-0.4-0.20.00.20.41234570 75 80 85 90 95 00Residual Actual FittedActual, Fitted and Residual Values from the Regressionof the Logarithm of General Fund Expenditures ($B) on YearNov. 2, 2000 ECON 240A-5 L. PhillipsMidtermc. 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?TableVariable Coefficient Std. Error t-Statistic Prob. C 0.011861 0.008030 1.477109 0.1465INDEX 0.776969 0.277141 2.803516 0.0074R-squared 0.145929 Mean 0.023018Adjusted R-squared 0.127362 S.D. 0.051722S.E.
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