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'Professor Mumford [email protected] edu Econ 360 †F whole 2012 conundrum Set 1 services True/ counterfeit (30 points) 1. FALSE If (ai , bi ) : i = 1, 2, . . . , n and (xi , yi ) : i = 1, 2, · · · , n atomic build 18 sets of n pairs of numbers, and so: n n n (ai xi + bi yi ) = i=1 i=1 ai x i + i=1 bi yi 2. FALSE If xi : i = 1, 2, . . . , n is a set of n numbers, past: n n n n n (xi ? x) = ? i=1 n i=1 2 x2 i ? 2? x i=1 xi + i=1 x = ? i=1 2 x2 ? n? 2 x i where x = ? 1 n i=1 xi 3. true up If xi : i = 1, 2, . . . , n is a set of n numbers and a is a constant, because: n n a xi = a i=1 n i=1 xi = a n x ? here x = ? 1 n i=1 xi 4. FALSE If X and Y be independent random uncertains so: E (Y |X) = E (Y ) 1 5. TRUE If {a1 , a2 , . . . , an } be constants and {X1 , X2 , . . . , Xn } atomic number 18 random variables then: n n E i=1 ai X i = i=1 ai E (Xi ) 6. FALSE For a random variable X, let µ = E (X). The variance of X can be expressed as: V ar(X) = E X 2 ? à ‚µ2 7. TRUE For random variables Y and X, the variance of Y conditional on X = x is given by: V ar(Y |X = x) = E Y 2 |x ? [E (Y |x)]2 8. TRUE An estimator, W , of ? is an unbiased estimator if E (W ) = ? for all possible apprizes of ?. 9.FALSE The central determine theorem verbalises that the comely from a random taste for any population (with ? nite variance) when it is standardized, by subtracting the guess and then dividing by the standard deviation, has an asymptotic standard normal distribution. 10. TRUE The faithfulness of large numbers states that if X1 , X2 , . . . , Xn are independent, identically distributed random variables with mean µ, then ? plim Xn = µ 2 Multiple Choice Questions (a) ceteris paribus (b) correlation coefficient (c) causative e? ect (d) independence (20 points) 11. The nous of holding â€Å"all else equal” is know as 12.If our entropyset has one observation for all(prenominal) state for the year 2000, then our informationset is (a) cross-section(a) data (b) pooled cross-sectional data (c) cartridge holder series data (d) panel data 13. If our dataset has one observation for every state for the year 2000 and an different observation for all(prenominal) state in 2005, then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) clipping series data (d) panel data 14. If our dataset has one observation for the state of atomic number 49 each year from 1950-2005 then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) term series data (d) panel data 15.Consider the function f (X, Y ) = (aX + bY )2 . What is (a) 2aX (b) a(aX + bY ) (c) 2a(aX + bY ) (d) a2 X ? f (X,Y ) ? X 3 Long Answer Questions (50 points) 16. The sum of squared deviations (subtracting the average value of x from each observation on x) is the sum of the squared xi shun n times the square of x. in that location are ? several ship canal to describe this, here is one: n n xi (xi ? x) ? i=1 = i=1 n (xi ? x + x) (xi ? x) ? ? ? n = i=1 n (xi ? x) (xi ? x) + ? ? i=1 n x (xi ? x) ? ? = i=1 (xi ? x)2 + x ? ? i=1 n (xi ? x) ? (xi ? x) = 0, so ? and we know that i=1 n i=1 (xi ? x)2 ? 17. There are several ways to show that this expression equals the judge covariance mingled with x and y, here is one: n n xi (yi ? y ) ? i=1 = i=1 n (xi ? x + x) (yi ? y ) ? ? ? n = i=1 n (xi ? x) (yi ? y ) + x ? ? ? i=1 (yi ? y ) ? = i=1 (xi ? x) (yi ? y ) ? ? 18. Correlation and actor are not always the aforementioned(prenominal) thing. (a) A negative correlation meat that larger syndicate surface is associated with deject test performance. This could be because the relationship is causal meaning that having a larger class size actually hurts student performance.However, on that point are other(a) reasons we exponent ? nd a negative relationship. For example, children from more a? uent families might be more credibly to attend schools with smaller class sizes, and a? uent children generally score better on standardized tests. Another possibility is that inwardly a school, a principal might assign the better students to smaller classes. Or, well-nigh parents might insist that their children are in the smaller classes, and these same parents tend to be more involved in their children’s didactics. Given the potential for confounding factors such(prenominal) as these, ? ding a negative correlation between class size and test scores is not powerful evidence that smaller 4 class sizes actually lead to better performance. Thus, without other information, we cannot draw a meaningful stinting conclusions. A correct answer should explicate that we should be careful about conscription economic conclusions from simple correlations. (b) The pattern correlation between N and T is de? ned as: s rN T = N T sN sT where the sample covariance, sN T , is given by: sN T = 1 999 meter ? Ni ? N i=1 ? Ti ? T and the sample standard deviations are given by: sN = 1 999 1000 Ni ? N i=1 2 sT = 1 999 1000 ? Ti ? T i=1 2 Note that there are several alternative ways to pen this and statistical programs generally use other algorithms to calculate the correlation that are little prone to loss of precision referable to roundo? error or storage everyplace? ow. 19. Wage data (a) There are 526 observations. (b) There are 274 men in the sample. This means that the sample is 52. 09 percent male. (c) The average level of education in the sample is 12. 6 years. The median(a) level of education is 12 years. (d) The highest education level in the sample is 18 years of school. 9 people in the sample idea having 18 years of education. (e) The average periodic wage in the sample is $5. 90. The median hourly wage in the sample is $4. 65. 20. Fertility data (a) There are 363 women in the sample. (b) The average number of children ever born to a char in the sample is 2. 3. The median number is 2. (c) The largest number of children ever born to a woman in the sa mple 7. Six women say having seven children. (d) 25 percent of the sample lived in the eastern United States at age 16. (e) The average level of eduction in the sample is 13. 2 years. 5\r\n'