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Understanding Hypothesis Testing Fundamentals

Sep 20, 2024

Lecture Notes: Hypothesis Testing

Test of Hypothesis Overview

  • Importance: Used in statistics to estimate parameters or make decisions based on data.
  • Components:
    1. Null Hypothesis (H0): Assumes no effect or difference.
    2. Alternate Hypothesis (HA): Assumes an effect or difference exists.
    3. Test Statistics: Measure to decide between H0 and HA.
    4. P-value: Represents confidence in decision.
    5. Rejection Criteria: Decide conclusion based on rejection region.

Types of Tests

  • Two-tailed Test: Concerned if parameter is different from a certain value, regardless of direction.
    • E.g., HA: μ ≠ μ0
  • One-tailed Test: Concerned if parameter is greater or less than a certain value.
    • E.g., HA: μ > μ0 or μ < μ0

Sampling and Central Limit Theorem

  • Sample Mean (xÌ„) and Sample Standard Deviation (s) are calculated from a sample of size n.
  • Central Limit Theorem: If sample size is large, xÌ„ follows a normal distribution.
  • Test Statistic: z = (xÌ„ - μ) / (s / √n)

Rejection Region and Critical Values

  • Rejection Region: Based on critical values, decide acceptance or rejection of H0.
  • Critical Values: Threshold values for z at given confidence levels (e.g., 1.96 for 95% confidence in two-tailed test).

P-value Interpretation

  • P-value: Probability of observing test statistic at least as extreme as the one observed.
  • Confidence Levels:
    • p < 0.05: Results statistically significant.
    • p < 0.01: Results highly significant.
    • p > 0.05: Results not significant.

Example 1: Company Productivity

  • Scenario: Evaluate if productivity has changed (two-tailed test).
  • Given:
    • μ0 = 880 tons, xÌ„ = 871, s = 21, n = 50.
  • Calculation:
    • z = -3.03.
    • Conclusion: Reject H0 at both 5% and 1% significance levels.
    • P-value: 0.0024 (less than 0.01, highly significant).

Example 2: Sodium Intake

  • Scenario: Determine if daily intake > 3300 mg (one-tailed test).
  • Given:
    • μ0 = 3300 mg, xÌ„ = 3400 mg, s = 1100 mg, n = 100, α = 0.05.
  • Calculation:
    • z = 0.91, Zα = 1.645.
    • Conclusion: Not statistically significant, fail to reject H0.
    • P-value: 0.1814 (greater than 0.05, not significant).

Errors in Hypothesis Testing

  • Type I Error: Rejecting H0 when it's true.
  • Type II Error: Accepting H0 when it's false.

Note: Understanding p-values, confidence intervals, and errors is crucial for interpreting statistical results. Next lecture will expand on Type I and Type II errors.

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