Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Sampling Distributions
Let X1, ..., Xn be a random sample from N(μ, 1) distribution, where μ ∈ ℝ is unknown. In order to test H0 : μ = μ0 against H1 : μ > μ0, where μ0 ∈ ℝ is some specified constant, consider the following two tests:
(A) Reject H0 if and only if X̅n > c1, where c1 is such that \(P_{μ_0}\) (X̅n > c1) = α ∈ (0, 1) and X̅n = \(\frac{1}{n} \sum_{i=1}^n X_i\).
(B) Reject H0 if and only if Median {X1, ..., Xn} > c2, where c2 is such that \(P_{μ_0}\)(Median{X1, ..., Xn} > c2) = α ∈ (0, 1).
Then which of the following statements are true?
1
The test described in (A) is the uniformly most powerful test of size α
2
The test described in (B) is the uniformly most powerful test of size α
3
Pμ(X̅n > c1) → 1 as n → ∞ for all μ > μ0
4
Pμ0(Median{X1, ..., Xn} > μ0) = \(\frac{1}{2} \)