Questions -

Q1. Let {X_i : i = 1, 2, . . . , n} be independent Binomial(m_i, ρ) random variables with ρ ∈ (0, 1) and m_i a known, positive integer for each i. For example, X_i can be the number of employees participating in a pension plan at a firm with m_i employees. Then ρ is the participation rate in the entire population of employees.

(i) Let Y = _i=1∑ⁿ X_i. What are the mean and variance of Y?

(ii) The MGF for the Binomial(m, ρ) distribution can be shown to be [1 - ρ + ρ exp(t)]^m. Use this to show that Y has a binomial distribution and specify its two arguments.

(iii) Suppose you want to test H₀ : ρ < ρ₀ against H₁ : ρ > ρ₀ for some value p₀ ∈ (0, 1). Explain why it will not always be possible to choose a critical value to obtain an exact 5% test. (Hint: A simple example with a small n will do.)

(iv) Explain how to compute the p-value for the test if y is the observed value of the test statistic. (Of course you also know m = _i=1∑ⁿm_i.)

(v) Use the Stata function binomialtail (. , . , .) to compute the p-value when ρ₀ = 0.5, m = 600, and y = 324. Would you reject the null at the 5% level? At the 1% level?

Q2. Let {X_i : i - 1, . . . , n} be IID draws from the Exponential(μ) distribution for μ > 0.

(i) Let Y = X₁ + X₂ + · · · X_n. What is the distribution of Y?

(ii) Conclude that W = Y/ μ ∼ Gamma(n, 1). What are E(W) and Var(W)?

(iii) Suppose you want to test H₀: μ < μ₀ against H₁: μ > μ₀, for some μ₀ > 0. How would you use the statistic T = Y/ μ₀ to obtain a test with size 5% when μ = μ₀? Find the critical value when n = 24 and report it to three decimal places. [Hint: The invgammap in Stata will be useful.]

(iv) Argue that μ = μ₀ is the least favorable case under H₀.

(v) Instead of finding a critical value, how would you find the p-value if the outcome of T is t? Report the p-value to three decimal places if n = 24 and t = 31.4.

Q3. Let {X_ni : i = 1, 2, . . . ,n} be a sequence of independent, identically distributed Exponential(μ_n) random variables. Therefore, the density of each X_ni is

f_n(x) = (1/μ_n) exp(-x/μ_n), x > 0.

Assume specifically that μ_n = 2 + δ/√n where δ ≥ 0. Let X^-_n be the sample average of {X_ni : i = 1, . . . , n}.

(i) Find E(X^-_n) and Var(X^-_n) in terms of δ and n.

(ii) Show that X^-_n →^p 2.

(iii) Define

T_n1 = √n(X^-_n - 2)/2

T_n2 = √n(X^-n - 2)/X^-_n

What is the asymptotic distribution of T_n1 when δ = 0? What about T_n2? Justify your answers.

(iv) Explain how you would test H₀ : μ < 2 against H₁ : μ > 2 using both of the statistics.

Q4. Use the data in EARNS2.DTA, which is available on the D2L site, to answer the following questions. These data are on workers at a data entry company, where they are mainly paid based on output.

(i) Use the command sum earns 0 5 earns 0 6. How many workers are there in the sample? Did average earnings go up in 2006 compared with 2005? What is the percentage change in average earnings?

(ii) At the beginning of 2006 the company invested in faster computers with the hope that the workers could become more productive (finish more jobs in the same amount of time). The variable cearns is the change in earnings for each worker: cearns_i = earns06_i - earns05_i. Use the command count if cearns < 0 to find the number of workers whose earnings actually fell.

(iii) Let μ = E(cearns) be the expected change in earnings (defined for a large population of data entry workers). Report the p-value for testing H₀ : μ ≤ 0 against H₁ : μ > 0. Do you reject H₀ at the 5% significance level? What about at the 1% level? What are the degrees of freedom in the t distribution?

(iv) Provide an economic reason and a statistical reason for why this analysis might not be convincing.

Q5. Provide an answer to each of the following questions.

(i) State whether you agree or disagree with the following statement, and provide justification: "If we base a statistical test on an unbiased estimator, the test is also unbiased."

(ii) Suppose that you believe you are sampling from an Exponential(θ) distribution with E(X) = θ. Consider four test statistics for testing H₀ : θ ≤ θ₀ against H₁ :  θ > θ₀ for some θ₀ > θ:

T₁ = √n(X^- - θ₀)/X^-

T₂ = √n(X^- - θ₀)/S

T3 = √n(X^- - θ₀)/θ₀

T4 = √n(X^- - θ₀)/√(n^-1 _i=1Σⁿ(X_i-θ₀)²)

where X^- is the sample average and S is the sample standard deviation. Which of these statistics is robust to failure of the Exponential distribution? Justify your answer.