Lecture Notes: Hypothesis Testing with P-Values

Introduction

• Continuation of hypothesis testing from Chapter 10.
• Focus on hypothesis testing using P-values instead of classic test statistic vs. critical value method.

Key Concepts

• P-value vs. Significance Level
  • Using P-value to determine hypothesis test results.
  • P-value compared to significance level (Alpha) to evaluate hypotheses.

Examples

Example 1: Z Test Statistic

• Hypothesis:
  • Null hypothesis (H₀): μ = 70
  • Alternative hypothesis (H₁): μ > 70
• Z Test Statistic: 2.02
• Critical Value & Significance Level
  • Significance Level (Alpha): 0.05
  • Critical region: area corresponding to Alpha.
• Finding P-value
  • P-value is the area under the curve from test statistic to reject region.
  • P-value = Probability(Z > 2.02)
  • Using Z table, compute: P-value = 1 - 0.9783 = 0.0217
• Conclusion: P-value < Alpha; Reject H₀.

Example 2: T Test Statistic

• Hypothesis:
  • Null hypothesis (H₀): μ₁ - μ₂ = 2
  • Alternative hypothesis (H₁): μ₁ - μ₂ > 2
• T Test Statistic: 1.04
• Significance Level & Reject Region
  • Significance Level (Alpha): 0.05
  • One-sided upper tail test
• Finding P-value
  • Degrees of Freedom = 20
  • P-value = Probability(T > 1.04)
  • Using T table, approximate: P-value ≈ 0.15
• Conclusion: P-value > Alpha; Do not reject H₀.

Summary and Comparison

• Critical Value vs. Test Statistic
  • Critical value corresponds to significance level (Alpha).
  • Test statistic corresponds to P-value.
• Direct relationship:
  • Given a test statistic, you can find P-value and vice-versa.
  • Given a critical value, you can find Alpha and vice-versa.
• Conclusion Method:
  • Compare P-value to Alpha to determine reject/do not reject decision.

Additional Notes

• Proportion equations will be provided in class.
• Any questions will be addressed in class discussion.

This lecture provided an understanding of how P-values are used in hypothesis testing, contrasting it with the method of comparing test statistics to critical values. Examples illustrated the process of calculating and interpreting P-values to make informed decisions about hypotheses.