Various Types of H₀ and H₁

Case I

A simple null vs. a one-sided upper-tailed composit alternative: $H_{0}:\theta =\theta _{0};\qquad H_{1}:\theta \gt\theta _{0}$

Case II

A simple null vs. a one-sided lower-tailed composit alternative:

$H_{0}:\theta =\theta _{0};\qquad H_{1}:\theta <\theta _{0}$

Remark: Testing procedure is equivalent to Case I.

Case III

A composit null vs. a one-sided upper-tailed composit alternative:

$H_{0}:\theta \leq \theta _{0};\qquad H_{1}:\theta \gt\theta _{0}$

Remark: Testing procedure is exact opposite of Case I.

Case IV

A composit null vs. a one-sided lower-tailed composit alternative:

$H_{0}:\theta \geq \theta _{0};\qquad H_{1}:\theta <\theta _{0}$

Remark: Testing procedure is equivalent to Case III or exact opposite of Case I.

Case V

A simple null vs. a two-sided composit alternative:

$H_{0}:\theta =\theta _{0};\qquad H_{1}:\theta \neq \theta _{0}$

Remark: Testing procedure is the combination of Case I and III.