null and alternative hypotheses

The null hypothesis ([latex]H_0[/latex]) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

• One-sample null hypothesis: [latex]H_0: \mu=\mu_0[/latex], [latex]\mu_0[/latex] is the null value.
• Two-sample null hypothesis: [latex]H_0: \mu_1=\mu_2[/latex] or [latex]H_0: \mu_1-\mu_2=0[/latex]

The alternative hypothesis ([latex]H_A[/latex]) is a claim about the population that is contradictory to [latex]H_0[/latex] and what we conclude when we reject [latex]H_0[/latex].

• One-sample alternative hypothesis:
  • [latex]H_A: \mu \lt \mu_0[/latex], [latex]\mu_0[/latex] is the null value.
  • [latex]H_A: \mu>\mu_0[/latex], [latex]\mu_0[/latex] is the null value.
  • [latex]H_A: \mu\ne \mu_0[/latex], [latex]\mu_0[/latex] is the null value.
• Two-sample alternative hypothesis:
  • [latex]H_A: \mu_1\lt \mu_2[/latex] or [latex]H_A: \mu_1-\mu_2\lt 0[/latex]
  • [latex]H_A: \mu_1>\mu_2[/latex] or [latex]H_A: \mu_1-\mu_2>0[/latex]
  • [latex]H_A: \mu_1\ne \mu_2[/latex] or [latex]H_A: \mu_1-\mu_2\ne0[/latex]

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.