Understanding Student's T-Distribution

Oct 17, 2024

Confidence Rules for Means

Recap from Last Section

  • Discussed true proportions (e.g., proportion of females in college)
  • Focus was on quantitative variables

Current Focus

  • Dealing with averages of quantity variables
  • Answer questions like "What's the true mean or average age of students in college?"
  • Building confidence intervals for population mean ( \mu )

Sampling Distribution

  • Use sampling distribution for ( \mu )
  • Problem: Unknown population standard deviation

Solution: Student's T-Distribution

  • Used when population standard deviation is unknown
  • Graph Characteristics:
    • Different curves for different degrees of freedom (df)
    • Degrees of Freedom (df): n - 1, where n = sample size
    • Curves are bell-shaped but have different tails
  • As df increases, curves resemble standard normal distribution

Student's T-Distribution Table

  • Represents degrees of freedom and area in right tail
  • Example Calculations: Degree of Freedom (df):
    • If n = 10, df = 9
    • Curves approach standard normal distribution as df increases

T-Scores

  • Replace z-scores when using t-distribution
  • Finding T-Scores: Steps & Examples
    • Find intersection of df and area in one tail
    • If df is not directly in table, round to nearest df available

Example Problems

  • Example A:

    • df = 7, area to the right = 0.005
    • T-score = 3.449

  • Example B:

    • n = 15, df = 14, area to the left = 0.10
    • T-score = -1.345 (mirror image on left side)

  • Example C:

    • n = 20, df = 19, confidence level = 98%
    • T-scores: Right side = 2.539, Left side = -2.539

  • Example D:

    • n = 54, df = 53, confidence level = 90%
    • T-scores: Right side = 1.576, Left side = -1.576
    • Note: Use df closest to 53; used df = 50

Key Takeaways

  • T-distribution is used for small sample sizes or unknown population variance
  • Degree of freedom influences shape of t-distribution curve
  • Larger samples make the t-distribution curve closer to normal distribution
  • Confident interval estimation involves determining appropriate t-scores based on df and tail areas