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T-Distribution vs Normal Distribution Comparison

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About This MicroSim

"Let's crack this nut!" Sylvia adjusts her glasses excitedly. "The t-distribution is like the normal distribution's more cautious cousin. It says, 'Hey, since we had to estimate the standard deviation from our sample, let's be a bit more humble about our certainty.'"

This MicroSim helps you understand why we need the t-distribution when doing inference about means, and how it compares to the standard normal (Z) distribution.

How to Use

  • Degrees of Freedom Slider: Drag to change df from 1 to 100 and watch the t-distribution change shape
  • Confidence Level Buttons: Toggle between 90%, 95%, and 99% to see different critical values
  • Show Tail Areas: Toggle to visualize the probability in the tails
  • Show Critical Values: Toggle to see the vertical lines marking critical values

Key Insights

  • The t-distribution has heavier tails than the normal distribution
  • With small df (small samples), the t-distribution is noticeably different from normal
  • As df increases, the t-distribution approaches the normal distribution
  • At df = 30+, the distributions are quite similar; at df = 100+, nearly identical
  • Heavier tails mean larger critical values for the same confidence level
  • This translates to wider confidence intervals with small samples

Lesson Plan

Learning Objective

Students will compare the shapes of t-distributions with different degrees of freedom to the standard normal distribution, understanding how heavier tails affect inference (Bloom's Taxonomy: Understanding L2).

Warm-Up Activity (5 minutes)

Ask students: "If we're less certain about something, should our confidence interval be wider or narrower?" Discuss how uncertainty should translate to more caution in our estimates.

Exploration Activity (10 minutes)

  1. Start with df = 5 and observe how much fatter the tails are
  2. Slowly increase df and watch the curves converge
  3. Compare critical values at each stage
  4. Predict: At what df will the difference be less than 0.1?

Discussion Questions

  1. Why does estimating sigma add extra uncertainty?
  2. Why do we "lose" a degree of freedom when computing s?
  3. For a sample of size 15, which has the larger 95% critical value: z or t?
  4. What happens to the width of a confidence interval as sample size increases?

Assessment

Have students calculate the difference between t and z for df = 5, 20, and 50 at 95% confidence. They should explain why this matters for constructing confidence intervals.

Sylvia's Insight

"Those heavier tails aren't a bug, they're a feature! The t-distribution is being honest about our uncertainty. When we have a small sample and had to estimate the spread, it's only fair that we're more humble about what we know."