If a random sample of 50 customers is selected from a

Question

Econ 310, Spring 2014- Week 8:

Problem 1- {X₁, X₂, X₃, X₄} is a simple random sample with distribution N (2, 2²), and the sample mean is X¯ = ¼ _i=1∑⁴X_i. After drawing the random sample, we get the observation {-1, 0, 5, 3}.

1. What is E[X¯]? Will it change if you have a different sample, for example, {4, -1, 2, 6}?

2. What is V (X¯)? Will it change if you have a different sample?

3. What is the distribution of X¯? Do we have a enough information to answer this question?

4. Now, if Xi ∼ Exponential (λ = 0.5), what is the distribution of X¯ = ¼ _i=1∑⁴ X_i?

5. If Xi ∼ Exponential (λ = 0.5) still, but we get a random sample with n = 64, what is the distribution of X¯ = 1/64 _i=1∑⁴ X_i?

Problem 2- The amount of time a bank teller spends with each customer has a population mean µ = 3.10 minutes and standard deviation σ = 0.40 minute.

1. If a random sample of 50 customers is selected from a population of 500 customers, what is the probability that the average time spent per customer will be at least 3 minutes?

2. Now, suppose that we observe only 16 customers, and answer the same question.

Problem 3- In a binomial experiment {X_i; i ≤ n} with n = 300 and p = 0.55, we define a binomial random variable X = _i=1∑³⁰⁰X_i ∼ Binomial (300, 0.55), and the proportion of success pˆ = X/300.

1. Find the approximate probability that X = 165.

2. Find the approximate probability that pˆ is greater than 60%.

3. We would like to repeat the same binomial experiment with p = 0.55, but with the smaller sample size. If we want to use the normal approximation to analyze pˆ, at least how many sample we need?