Question

1. Design a hypothetical ideal randomized controlled experiment to study:
   • the effect of wearing seat belts on highway traffic deaths, &
   • the effect of hours spent studying on performance on economics exams
     Suggest some impediments to implementing these experiments in practice. (6 points)
2. The following table gives the joint probability distribution between employment status & college graduation in the US working age population for 2012.

   	Unemployed (Y=0)	Employed (Y=1)	Total
   Non-college graduates (X=0)	0.053	0.586	0.639
   College graduates (X=1)	0.015	0.346	0.361
   Total	0.068	0.932	1

   • Compute E(Y). (1 point)
   • Show that 1 - E(Y) is the unemployment rate. (the fraction of the labor force that is unemployed) (1 point)
   • Calculate E(Y|X=0) & E(Y|X=1). (1 point)
   • Calculate the unemployment rate for college graduates, & for non-college graduates. (1 point)
   • A randomly selected member of the population reports being unemployed. What is the probability that they are a college graduate? A non-college graduate? (2 points)
   • Are educational achievement & employment status independent? Explain. (1 point)
   • Does this relationship prove that college education has a causal effect on employment status? Use the potential outcomes framework to explain why or why not. (2 points)

Answer 1

a)

E(Y) = 0*P(Y=0) +1*P(Y=1)

E(Y) = 0.932

b)

1-E(Y)=0.068

but from table it seems to be probability of unemployment i.e. rate of unemployment=1-E(Y)=0.068

c)

E(Y|X=0) = E(Y|person is non college graduate)

E(Y|X=0) = 0*P(Y=0|X=0) +1*P(Y=1|X=0)

E(Y|X=0) = 0* 0.053+1* 0.586 =0.586

E(Y|X=0) = 0.586   

E(Y|X=1) = E(Y|person is college graduate)

E(Y|X=1) = 0*P(Y=0|X=1) +1*P(Y=1|X=1)

E(Y|X=0) = 0* 0.068+1* 0.932 =0.932

E(Y|X=0) = 0.932

d)

unemployment rate for college graduates = P(unemployment | person is college graduate)

unemployment rate for college graduates = P(person is unemployed and college graduate)/P(person is college graduate)

unemployment rate for college graduates = P(X=1,Y=0)/P(X=1) = 0.015/0.361 =0.04155

e)

unemployment rate for non college graduates = P(unemployment | person is non college graduate)

unemployment rate for non college graduates = P(person is unemployed and non college graduate)/P(person is non college graduate)

unemployment rate for non college graduates = P(X=0,Y=0)/P(X=0) = 0.053/0.639 =0.0829

f) A randomly selected member of the population reports being unemployed. Then the probability that they are a college graduate = P(X=1|Y=0) =P(X=1,Y=0)/P(Y=0)

=0.015/0.068 = 0.2206

A randomly selected member of the population reports being unemployed. Then the probability that they are a non college graduate = P(X=0|Y=0) =P(X=0,Y=0)/P(Y=0) = 1- P(X=1,Y=0)/P(Y=0) = 0.7794

g) P(X=0,Y=0)= 0.053

P(X=0)*P(Y=0) = 0.639*0.068 = 0.04345

therefore, P(X=0,Y=0) ≠ P(X=0)*P(Y=0)

hence educational achievement & employment status are not independent.