Hypothesis Testing and Errors

Actual Population Value

  • Refers to the true value of a parameter in the population being studied.

Hypotheses

  • Ho (Null Hypothesis): A statement that there is no effect or no difference, used as a default assumption in statistical testing.

  • H₁ (Alternative Hypothesis): This asserts the opposite of the null, indicating some effect or difference exists.

Decisions in Hypothesis Testing

  • Reject Ho: This means there is sufficient evidence to support the alternative hypothesis H₁.

  • Fail to Reject Ho: This suggests that there is not enough evidence to support H₁.

Types of Errors

  • Type I Error: Incorrectly rejecting the null hypothesis when it is true.

    • Notation: Probability of Type I Error is denoted as eta.

  • Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true.

    • Notation: Probability of Type II Error is P(Type ext{ }II ext{ }Error).

Power of a Test

  • Power: The probability that a test correctly rejects a false null hypothesis (i.e., the test correctly identifies a true effect).

    • Formula: P(Type ext{ }I ext{ }Error) = 1 - ext{Power}

Factors that Increase Power

  1. Increase Sample Size: Larger samples provide more information, leading to more accurate estimates.

  2. Increase Significance Level (\alpha): Raising \alpha decreases the criteria for rejecting the null hypothesis.

  3. Decrease Standard Error: Lower variability leads to more precise estimates, enhancing the detection of effects.

  4. True Parameter Value is Farther from the Null: When the true effect is larger, it is easier to detect in hypothesis tests.