ec41f10_hw5_sol

ec41f10_hw5_sol - Kata Bognar kbognar@ucla.edu Economics 41...

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Kata Bognar kbognar@ucla.edu Economics 41 Statistics for Economists UCLA Fall 2010 Homework Assignment 5. - suggested solutions by Yujing Xu - NOTE: Please show your calculations for Questions 1-2, 6-13 and 16-19. The exercises are from the textbook (Tanis and Hogg: A Brief Course in Mathematical Statistics). Distribution of the sample mean. 1. The table below provides the wealth of the world’s six richest people in 2003. Consider these six people a population of interest. Person Wealth ($ billions) William H. Gates III (G) 41 Warren E. Buffett (B) 31 26 Paul G. Allen (P) 20 Prince Alwaleed Bin Talal Alsaud (T) 18 Lawrence J. Ellison (E) 17 (a) Calculate the mean wealth of the six people, μ . Answer: μ = (41 + 31 + 26 + 20 + 18 + 17) / 6 = 153 / 6 = 25 . 5 (b) List all the possible samples of size 2 from this population. Calculate the mean wealth for all possible samples. Denote ¯ X 2 the sample mean wealth for samples of size 2. Derive the distribution of ¯ X 2 . What is the expected value of ¯ X 2 ? Answer: The possible samples of size 2 are GB,GA,GP,GT,GE,BA,BP,BT,BE,AP,AT,AE,PT,PE,TE Please refer to the following table for the sample mean wealth for these samples. mean wealth G B A P T E G 36 33.5 30.5 29.5 29 B 28.5 25.5 24.5 24 A 23 22 21.5 P 19 18.5 T 17.5 The sample mean for samples of size 2 assumes the values above with equal probabilities of P x j ) = 1 / 15 . The expected value of the sample mean for samples of size 2 is E ( ¯ X 2 ) = j ¯ x j / 15 = 25 . 5 . Notice that E ¯ X 2 = μ. (c) List all the possible samples of size 4 from this population. Calculate the mean wealth for all possible samples. Denote ¯ X 4 the sample mean wealth for samples of size 4. Derive the distribution of ¯ X 4 . What is the expected value of ¯ X 4 ? Answer: The possible samples of size 4 are APTE,BPTE,BATE,BAPE,BGAP,GPTE,GATE,GAPE,GAPT,GBTE,GBPE,GBPT, BGAE,BGAT,GBAT . The sample mean wealth for all possible samples are shown in the following table. In the table, entry (X,Y) means the mean wealth of the sample { G,B,A,P,T,E } / { X,Y } , where X,Y can be G,B,A,P,T,E . For example, entry (G,B) means the mean wealth of sample { A,P,T,E } mean wealth G B A P T E G 20.25 21.5 23 23.5 23.75 B 24 25.5 26 26.25 A 26.75 27.25 27.5 P 28.75 29 T 29.5 1
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The sample mean for samples of size 4 assumes the values above with equal probabilities of P x j ) = 1 / 15 . The expected value of the sample mean for samples of size 4 is E ( ¯ X 4 ) = j ¯ x j / 15 = 25 . 5 . Notice that E ¯ X 4 = μ. (d) Compare the distributions in part (b) and (c). Answer: Both distributions are centered around the population mean, i.e. E ( ¯ X 2 ) = E ( ¯ X 4 ) = μ. However the distribution of ¯ X 4 is more concentrated around that value. You can see this if you plot the values and/or looking at the range of the values for both random variables. Alternatively you can calculate the standard deviation for both random variables to reach the same conclusion. 2. According to the U.S. Census Bureau, the mean price of new mobile homes is $61,300. The
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ec41f10_hw5_sol - Kata Bognar kbognar@ucla.edu Economics 41...

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