s07pre1a - Prof. G. Jakubson Economics 321 Spring 2007...

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Prof. G. Jakubson Economics 321 Spring 2007 PRELIMINARY EXAM #1 - ANSWER KEY 1. Class intervals for infant mortality rate Countries cum rel freq rel freq freq 0 to <10 24 24/131 = .18 24/131 = .18 10 to < 20 24 24/131 = .18 48/131 = .36 20 to <30 14 14/131 = .11 62/131 = .47 30 to <50 18 18/131 = .14 80/131 = .61 50 to <75 18 18/131 = .14 98/131 = .75 75 to <100 16 16/131 = .12 114/131 = .87 100 to <140 13 13/131 = .10 127/131 = .97 140 to 180 4 4/131 = .03 131/131 = 1 Total number of countries 131 a. Median infant mortality rate is in the 30 to < 50 category. Since we were asked for the median and not just the class, we must interpolate: Median = 30 + [.5 - .47)/.14]*(50-30) = 34.3. This assumes a uniform distribution within the class. b. See above. c. Use grouped mean and variance expressions: Mean = xf x n n k k k k k k ∑∑ = and Variance = n n xx f k k k 1 2 () , where x k is the class mark and f k is relative frequency. 2. a. [] Z= 1 n ∑− =∑ −∑=−= XX s n X sn X s X s X s ix i x x / 11 0 or use the linear transformation formulae: Z i = a + bX i where x x s 1 b and s X - a = = b. s z = (1/s x ) s x = 1. c. Since T i is just a standardized version of Y i , its mean and variance are also 0 and 1, respectively. d. Variance of ee . Here, X=log(food expenditure). Evaluated at s x 2 2 X , this yields . Therefore, the std of e . . e 821 2 2 46 2861746 = X 2861746 =1692. Note that this approximates the std of food expenditure. In order to compare this to the standard deviation of F from the table, we must adjust the units. Log(food expenditure) (LF) uses food expenditure data in $1983, while food expenditure (F) uses data in 1,000’s of $1983. So, we must divide the std we got by 1000 (=1.692) to put it into $1983. Compare 1.692 to the std of F from the table, which is 1.85. The approximation is about 9% lower than the actual.
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s07pre1a - Prof. G. Jakubson Economics 321 Spring 2007...

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