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Null, alternative hypothesis; errors

Short answer

Null hypothesis states no difference or effect and uses equality; alternative states a difference or direction. Type I error (α) is rejecting a true H0; Type II error (β) is accepting a false H0. Decreasing α increases β unless sample size grows.

Explain Null hypothesis, alternative hypothesis and the errors arising in the test of hypothesis.

Null hypothesis (H0)

The statement of no difference or no effect that is tested, and is assumed true until the sample evidence rejects it. It always contains an equality sign. Example: H0: μ = 50 (mean life is 50 hours).

Alternative hypothesis (H1 or Ha)

The statement that is accepted if H0 is rejected. It says there is a difference or effect. It decides the type of test:

  • H1: μ ≠ 50: two-tailed test
  • H1: μ > 50: right-tailed test
  • H1: μ < 50: left-tailed test

Errors in testing

The decision rests on sample data, so it can be wrong.

DecisionH0 trueH0 false
Reject H0Type I error, probability αCorrect decision (power 1 − β)
Accept H0Correct decision (1 − α)Type II error, probability β
  • Type I error: rejecting H0 when it is actually true. Its probability is the level of significance α (e.g. 0.05). Example: rejecting a good lot.
  • Type II error: accepting H0 when it is actually false. Its probability is β. Example: accepting a bad lot.
  • For a fixed sample size, decreasing α increases β. Both can be reduced only by increasing the sample size.

✓ Answer checked by Claude Sonnet 5.5

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