Question

b) Assume real unemployment rate is 5%. Look generally at contacting N people. ?_? of those N people states that they are
unemployed. We use the estimator ?_? = ?_? / ? to estimate unemployment rate. Which
distribution follows ?_? when N becomes large and why? Find the expectation E(?_?) and variance of ?_?.
My answer is: It follows Normal distribution because of the Central Limit Theorem, which states that ?_? will follow standard normal distribution
when N becomes very large.

. Expectation is: np/n = p = 0,05

Variance is: np(1-p)/n = p(1-p) = 0,05 * 0,95 = 0,0475

Is this correct answer?

c) What is the probability of observing ?_? ≥ 0.07 when we contact 20, 50, 100, and 1000 people if you use the results from b)?

Answer 1

Your expectation is correct but the variance is in correct.

The variance is given as:

VAR(RN)=Var(AN/N)=Var(AN)/N²=Np(1-p)/N²=p(1-p)/N

Var(RN)=0.05*(1-0.05)/N²=0.0475/N²

Therefore

When N=20

When N=50

When N=100

When N=1000