Lecture on Type Two Error and Power of a Test

Definitions

• Type Two Error (β): Probability of failing to reject the null hypothesis (H₀) when it is false.
• Power of a Test: Probability of correctly rejecting the null hypothesis when a specific alternative hypothesis is true. Calculated as 1 - β.

Key Concepts

• The power of the test is a measure of the test's ability to detect an effect when there is one.
• Power is 1 minus the Type Two Error probability.
• High power (close to 1) is desirable as it indicates a strong test.

Example Problem

• Scenario: Weights of males at a college.
• Objective: Find the Type Two Error when the actual mean (μ) is 70, with critical values at 67 and 69.

Process

1. Original Curve (H₀):
   • Critical regions are at 67 and 69.
2. Alternative Curve (H₁):
   • Under this curve, determine probability of not rejecting H₀.
   • This involves calculating where the actual mean is 70.
3. Z Values Calculation:
   • For the sample size n = 64 and σ = 3.6:
   • Z₁ = (67 - 70) / (3.6 / √64) = -6.67
   • Z₂ = (69 - 70) / (3.6 / √64) = -2.22
4. Probability Calculation:
   • P(Z < 2.22) - P(Z < 6.67) = 0.0132 - 0 = 0.0132 (β)
5. Power of the Test:
   • 1 - β = 0.987 (strong power)

Additional Example

• Objective: Find Type Two Error when μ = 68.5.

Process

1. Z Values Calculation:
   • Z₁ = (67 - 68.5) / (3.6 / √64) = -3.33
   • Z₂ = (69 - 68.5) / (3.6 / √64) = 1.11
2. Probability Calculation:
   • P(Z < 1.11) - P(Z < -3.33) = 0.8665 - 0.00004 = 0.8661 (β)
3. Power of the Test:
   • 1 - β = 0.1349 (weak power)
   • More likely to detect larger differences in means.

Conclusion

• Key Insight:
  • Larger sample sizes increase the probability of detecting smaller differences between true and hypothesized means.
  • Dedicate extra time to understanding Type Two Error and Power due to the complexity of overlapping curves and calculations.