t-distribution

t-Distribution

When working with small samples or when the population standard deviation is unknown, we use the t-distribution.

Key Properties

• Symmetric around zero
• Bell-shaped like the Normal Distribution
• Characterized by degrees of freedom (df = $n-1$)
• Has heavier tails than the Normal distribution
• Approaches the Normal distribution as sample size increases ($n\to \mathrm{\infty }$)

Comparing Normal and t-Distributions

Confidence Intervals

Using Normal Distribution ($n\ge 30$)

For 95% confidence intervals when $\sigma$ is known or $n\ge 30$:

Using t-Distribution (Small Samples)

For small samples or unknown $\sigma$:

where $t_{\alpha /2,n-1}$ is the critical value from t-distribution with $n-1$ degrees of freedom

Effect of Parameters on Intervals

The width of confidence intervals depends on:

• Confidence Level: Higher → Wider interval
• Sample Size: Larger → Narrower interval
• Variability: More variable → Wider interval

Conditions for Valid Application

For Normal Distribution

1. Random sample from the population
2. Sample size $n\ge 30$

For t-Distribution

1. Random sample from the population
2. For small $n$: Population should be approximately normal
3. Independent observations

When to Use Each

• Use t-distribution when:
  • Sample size is small ($n<30$)
  • Population standard deviation ($\sigma$) is unknown
  • Working with differences in means
• Use Normal distribution when:
  • Sample size is large ($n\ge 30$)
  • Population standard deviation is known
  • Working with proportions

See also:

• Confidence Interval
• Standard Error
• degrees of freedom