P-Value and Hypothesis Testing - Chapter 8 Notes

P-Value and Hypothesis Testing - Chapter 8 Notes

Key Concepts of Hypothesis Testing

  • Understanding Hypothesis Testing: A statistical method using sample data to evaluate the validity of a hypothesis about a population.
  • Foundational Steps:
  • State null (H0) and alternative hypotheses (H1).
  • Set criteria for decision based on critical regions.
  • Conduct tests (e.g., z-test) on collected data.
  • Make a probability-based decision to reject or retain the null hypothesis.
  • Errors:
  • Type I Error: Rejecting a true null hypothesis.
  • Type II Error: Failing to reject a false null hypothesis.

Hypotheses

  • Definition: A precise, testable prediction about an event or outcome.
  • Null Hypothesis (H0): No meaningful change or effect at the population level (e.g., H0: μControl = μTreatment).
  • Alternative Hypothesis (H1): States there is a significant effect/change (e.g., H1: μControl ≠ μTreatment for two-tailed tests or H1: μControl > μTreatment for one-tailed tests).

Logic of Hypothesis Testing

  1. State Hypothesis: Predict about the population.
  2. Predict Characteristics: Based on hypotheses, forecast what the sample data should look like.
  3. Collection of Data: Gather a random sample.
  4. Comparison: Evaluate the sample data against the predictions from the hypothesis.
  • If consistent, retain H0; if inconsistent, reject H0.

Steps in Hypothesis Testing

  1. State the Hypotheses:
  • Null Hypothesis (H0) and Alternative Hypothesis (H1).
  1. Set Criteria for Decision:
  • Identify critical regions where the null is rejected via alpha levels.
  • Common α-level is 0.05 (5%).
  1. Collect Data:
  • Obtain random samples and calculate the sample statistics, especially the mean.
  1. Make a Decision:
  • Compare the sample outcome to the critical value.
  • If z-score is in the critical region, reject H0; if not, fail to reject H0.

Decision Criteria and Alpha Levels

  • The Alpha Level (α): The probability threshold for rejecting the null hypothesis.
  • Critical regions are defined as areas where if the sample mean falls, H0 is rejected (e.g., α = 0.05 splits into two tails for two-sided tests: 0.025 in each tail).
  • Z-scores corresponding to different alpha levels are:
  • α = .05: z = ±1.96
  • α = .10: z = ±1.645
  • α = .01: z = ±2.576

Examples of Hypothesis Testing

  1. Example of Violence in Media:
  • H0: Children who view violent media show equal or lesser aggression than those who do not.
  • H1: Children who view violent media show greater aggression.
  1. Example of Note-Taking Method:
  • H0: μLaptop = μHand
  • H1: μLaptop ≠ μHand
  1. Example of Antidepressants:
  • H0: μNo Prozac ≤ μProzac
  • H1: μNo Prozac > μProzac

Interpretation of Results

  • If results fall within the critical region:
  • Reject H0 (suggests the effect is statistically significant).
  • If results do not fall within the critical region:
  • Fail to reject H0 (no significant evidence found).
  • Strive to use probability language when discussing results (e.g., p < α implies significant findings).

Summary of Hypothesis Testing Steps

  • Step 1: Formulate hypotheses, clarify if one- or two-sided.
  • Step 2: Determine critical regions based on chosen alpha level.
  • Step 3: Collect data and compute statistics (e.g., z-scores, p-values).
  • Step 4: Draw conclusions based on statistical significance to accept or reject the null hypothesis (H0).