Answer Key
Professor Mumford mumford@purdue.edu
Econ 360 - Fall 2012
Problem Set 1 Answers
True/False
(30 points)
1. FALSE If (ai , bi ) : i = 1, 2, . . . , n and (xi , yi ) : i = 1, 2, · · · , n are sets of n pairs of numbers, then: 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, then: 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 If xi : i = 1, 2, . . . , n is a set of n numbers and a is a constant, then: n n
a xi = a i=1 n i=1
xi = a n x ¯
where x = ¯
1 n i=1
xi
4. FALSE If X and Y are independent random variables then: E (Y |X) = E (Y )
1
5. TRUE If {a1 , a2 , . . . , an } are constants and {X1 , X2 , . . . , Xn } are 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 − 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 θ is an unbiased estimator if E (W ) = θ for all possible values of θ. 9. FALSE The central limit theorem states that the average from a random sample for any population (with finite variance) when it is standardized, by subtracting the mean and then dividing by the standard deviation, has an asymptotic standard normal distribution. 10. TRUE The law 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 (c) causal effect (d) independence
(20 points)
11. The idea of holding “all else equal” is known as
12. If our dataset has one observation for every state for the year 2000, then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) time series data (d) panel data 13. If our dataset has one
Econ 360 - Fall 2012
Problem Set 1 Answers
True/False
(30 points)
1. FALSE If (ai , bi ) : i = 1, 2, . . . , n and (xi , yi ) : i = 1, 2, · · · , n are sets of n pairs of numbers, then: 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, then: 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 If xi : i = 1, 2, . . . , n is a set of n numbers and a is a constant, then: n n
a xi = a i=1 n i=1
xi = a n x ¯
where x = ¯
1 n i=1
xi
4. FALSE If X and Y are independent random variables then: E (Y |X) = E (Y )
1
5. TRUE If {a1 , a2 , . . . , an } are constants and {X1 , X2 , . . . , Xn } are 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 − 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 θ is an unbiased estimator if E (W ) = θ for all possible values of θ. 9. FALSE The central limit theorem states that the average from a random sample for any population (with finite variance) when it is standardized, by subtracting the mean and then dividing by the standard deviation, has an asymptotic standard normal distribution. 10. TRUE The law 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 (c) causal effect (d) independence
(20 points)
11. The idea of holding “all else equal” is known as
12. If our dataset has one observation for every state for the year 2000, then our dataset is (a) cross-sectional data (b) pooled cross-sectional data (c) time series data (d) panel data 13. If our dataset has one