Lecture Notes: Using the T Distribution for Confidence Intervals

Introduction

• When dealing with certain statistical problems, you use the T distribution instead of the normal distribution.
• The process is similar, but you substitute:
  • s for Sigma
  • t for Z
• Everything else remains the same.

Problem Example

• Find a 95% confidence interval given:
  • Sample mean (X̄): 124
  • Standard deviation (s): 8.6
  • Sample size (n): 81

Steps to Solve

1. Write Down the Formula

• Formula: X̄ ± (t * s/√n)
  • Find each component:
    • X̄ is given: 124
    • Calculate s/√n:
      • s/√n = 8.6/√81 = 0.9556*

2. Check Distribution

• Use the T distribution instead of the normal distribution.
• Always ensure the correct distribution is applied.

3. Find the T Score

• Degrees of Freedom (df): n - 1 = 80
• Level of Significance: 0.025 (for two-tailed test)
• Tools to find t-score:
  • Stat Disk: Enter degrees of freedom and area to the right.
  • Excel Formula: T.INV.2T(0.05, 80)
  • Resulting T score: 1.99 (consider both + and - values)

4. Calculate Confidence Interval

• Substitute values into the formula:
  • Calculate t * (s/√n):
    • 1.99 * 0.9556 = 1.9016
  • Find upper and lower bounds:
    • Lower: 124 - 1.9016 = 122.0984
    • Upper: 124 + 1.9016 = 125.9016
• 95% Confidence Interval: [122.0984, 125.9016]

Conclusion

• The T distribution calculation for confidence intervals is straightforward if you follow the steps systematically.
• Always check whether the normal or T distribution is appropriate.
• Ensure correct formula application and verification.

Questions

• The floor was opened for questions regarding the T distribution.