Question

Suppose that X₁,….., X_n forms a random sample from the normal distribution with unknown mean µ and unknown variance σ². Classify each of the following hypotheses as either simple or composite:

a) H₀ : µ = 0 and σ=1

b) H₀ : µ > 3 and σ<1

c) H₀ : µ = -2 and σ²<5

d) H₀ : µ = 0

Short answer

a.Given hypothesis is a simple hypothesis.

b.Given hypothesis is a composite hypothesis.

c.Given hypothesis is a composite hypothesis.

d.Given hypothesis is a composite hypothesis.

Step-by-step solution

Given Information

The parameter space for the parameters of N( µ, σ² ) is given by R×R⁺ (as both the mean and variance are unknown). A hypothesis is simple if it contains a single point, and composite otherwise.

Determine if the following assumptions are simple or composite

a. Given hypothesis is H₀ : µ = 0 and σ=1

The space corresponding to the hypothesis is {(0,1)}. Thus this hypothesis is a simple hypothesis.

b. Given hypothesis isH₀ : µ > 3 and σ<1

The space corresponding to the hypothesis is (3,∞) × (0,1). Hence clearly this hypothesis is a composite hypothesis.

c. Given hypothesis isH₀ : µ = -2 and σ²<5

The space corresponding to this hypothesis is {-2} ×(0,√5).Hence clearly this hypothesis is a composite hypothesis.

d. Given hypothesis isH₀ : µ = 0. The space corresponding to this hypothesis is given by{0} × R⁺. Hence clearly this hypothesis is a composite hypothesis.