Lecture 9 slides: Testing concepts

Example For example, let X 2 1, ..., Xn be a random sample from distribution N(μ, σ ) with σ known and μ ∈ M. Suppose our hypothesis is that μ ∈ M1 for some M1 ⊂ M, i.e. M1 is some subset of M. It is called the null hypothesis. It is denoted as H0 : μ ∈ M1. Then the alternative hypothesis is that μ ∈/ M1, i.e. μ ∈ M\M1. It is denoted as Ha : μ ∈/ M\M1. For example, if M1 = μ0 and M = {μ : μ ≥ μ0}, then H0 : μ = μ0 and Ha : μ > μ0. Or, as another example, ifM1 = μ0 andM = {μ : μ ∈ R}, then H0 : μ = μ0 and Ha : μ = μ0. If a hypothesis includes only one parameter value, it is called simple. Otherwise, the hypothesis is called complex. In the examples above, the null hypothesis was simple while the alternative was complex. In principle, we can also allow for complex null hypotheses. For example, ifM1 = μ1∪μ2 andM = {μ : μ ∈ R}, then H0 : μ = μ1 orμ = μ2 and Ha : μ = μ1 andμ = μ2. It is customary to mention both the null and the alternative hypotheses since the full parameter space M is often unspeci ed.