Lecture Notes: Type I and Type II Errors

Introduction to Type I and Type II Errors

• Type I Error: Occurs when the null hypothesis (H₀) is rejected when it is actually true.
  • Also known as a "false positive."
  • Represented by Alpha (α).
• Type II Error: Occurs when the null hypothesis is not rejected when it is false.
  • Also known as a "false negative."
  • Represented by Beta (β).

Relationship and Significance

• Alpha (α): Same alpha found in confidence levels (e.g., 1-α confidence).
• Example Context: Fire alarm scenario.
  • Type I Error: Alarm goes off when there is no fire.
  • Type II Error: No alarm when there is a fire.

Example Problem

• Hypothesis about average weight (mu, μ) of male students: H₀: μ = 68 kg vs. H₁: μ ≠ 68 kg.
• Critical Region: x̄ < 67 or x̄ > 69 (arbitrarily chosen).
• Normal Distribution: Mean set at 68.
• Type I Error Probability:
  • Use Z table for probabilities under normal curve.
  • Calculate Z values for critical regions:
    • Z₁ for x̄ = 67
    • Z₂ for x̄ = 69
  • Alpha (α) is calculated as 2 times the probability of Z < 1.67.
  • Result: Probability of Type I Error is 9.5%.

Properties of Type I and Type II Errors

1. Inverse Relationship: Decrease in one increases the other.
2. Critical Region Adjustment: Changing critical values affects Type I Error.
3. Sample Size Impact: Increasing sample size (n) reduces both α and β simultaneously.
   • More samples lead to more reliable results.
4. Effect of True Value:
   • If true parameter value approaches hypothesized value, β is maximum.
   • Greater distance between true and hypothesized values results in smaller β and better detection.

Conclusion

• Understanding the real-world context of Type I and Type II errors is crucial.
• Future videos will delve deeper into Type II errors.