t-Distribution

The Need for the t-Distribution

When we don’t know the population standard deviation (σ), we must estimate it from our sample (s). This estimation introduces extra uncertainty.

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What is the t-Distribution?

The t-distribution (also called Student’s t) is similar to the normal distribution but with heavier tails.

Key Properties

Property	Description
Shape	Symmetric, bell-shaped
Center	Mean = 0
Spread	More spread than normal (heavier tails)
Parameter	Degrees of freedom (df)

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Degrees of Freedom

The degrees of freedom (df) control the shape of the t-distribution.

Effect of df on Shape

df	Shape
Small (e.g., 3)	Very heavy tails, flat peak
Medium (e.g., 10)	Moderately heavy tails
Large (e.g., 30+)	Nearly normal
df → ∞	Exactly normal

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t vs z Distribution

Aspect	z (Normal)	t (Student’s t)
Use when	σ is known	σ is unknown (use s)
Tails	Lighter	Heavier (more extreme values likely)
Shape	Fixed	Changes with df
Critical values	Always same	Depend on df
Large samples	Use z	Approaches z

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Reading the t-Table

A t-table provides critical values for different:

• Degrees of freedom (rows)
• Tail probabilities (columns)

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Confidence Intervals with t

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Hypothesis Testing with t

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When to Use t vs z

Flowchart

1. Are you working with means or proportions?
   • Proportions → use z
   • Means → continue
2. Do you know population σ?
   • Yes → use z (rare)
   • No → use t

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Assumptions for t-Procedures

Sample Size	Normality Requirement
n < 15	Data must be nearly normal
15 ≤ n < 30	No strong skewness or outliers
n ≥ 30	OK even if somewhat skewed

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The History of “Student’s t”

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Summary

In this lesson, you learned:

• t-distribution is used when σ is unknown (we estimate with s)
• Has heavier tails than normal—accounts for extra uncertainty
• Shape depends on degrees of freedom (df = n - 1)
• As df increases, t approaches normal (z)
• Use t-critical values for confidence intervals and hypothesis tests
• Always valid for inference about means when σ is unknown
• Requires random sample and approximate normality (or large n)

Practice Problems

1. A sample of n = 12 has mean = 45 and s = 6. What df should you use for a t-procedure?

2. Find the t-critical value for a 90% CI with n = 20. (Hint: df = 19, look up two-tailed α = 0.10)

3. For n = 8, mean = 52, s = 4, construct a 95% CI for μ.

4. Why are t-based confidence intervals wider than z-based intervals (for the same confidence level)?

Click to see answers

1. df = n - 1 = 12 - 1 = 11

2. df = 19, two-tailed 90% means α/2 = 0.05 in each tail t* = 1.729 (from t-table)

3. df = 8 - 1 = 7 t* for 95% CI, df = 7: 2.365

SE = s/√n = 4/√8 = 1.414 ME = 2.365 × 1.414 = 3.34

CI: 52 ± 3.34 = (48.66, 55.34)

4. t-critical values are larger than z-critical values (e.g., t* = 2.26 vs z* = 1.96 for n = 10).

This is because:

• We’re estimating σ with s, which introduces extra uncertainty
• The t-distribution has heavier tails to account for this
• Wider intervals reflect our increased uncertainty

Next Steps

Apply t-distributions in hypothesis testing:

• t-Tests - One-sample, two-sample, paired
• Confidence Intervals - Complete CI methods
• T-Test Calculator - Practice calculations