Student's t-Distribution

Overview

The t-distribution is used when making inferences about population means when the population standard deviation is unknown and must be estimated from the sample.

When to Use

Use t-distribution instead of z when:

• Population standard deviation $\sigma$ is unknown
• Using sample standard deviation $s$
• Sample size is small (though valid for any $n$)

Degrees of Freedom

$$df = n - 1$$

Where $n$ is the sample size.

Properties

Property	t-Distribution
Shape	Symmetric, bell-shaped
Mean	0
Variance	$\frac{df}{df - 2}$ for $df > 2$
Tails	Heavier than normal
As $df \to \infty$	Approaches standard normal

Comparison to Normal

         Normal
           ↓
     ╭─────────╮
    ╱     t     ╲
───╱─────────────╲───

• t has heavier tails
• More probability in the tails
• Same center (mean = 0)

Key t-Values

$df$	$t_{0.10}$	$t_{0.05}$	$t_{0.025}$	$t_{0.01}$	$t_{0.005}$
1	3.078	6.314	12.706	31.821	63.657
5	1.476	2.015	2.571	3.365	4.032
10	1.372	1.812	2.228	2.764	3.169
20	1.325	1.725	2.086	2.528	2.845
30	1.310	1.697	2.042	2.457	2.750
$\infty$	1.282	1.645	1.960	2.326	2.576

Note: $t_\infty$ equals z-values (standard normal).

Notation

$$t_{\alpha, df}$$

Example: $t_{0.025, 10} = 2.228$ (two-tailed 95% critical value with $df = 10$)

T-Statistic for Sample Mean

$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$

Where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean
• $s$ = sample standard deviation
• $n$ = sample size

Applications

Confidence Interval for Mean

$$\bar{x} \pm t_{\alpha/2, df} \times \frac{s}{\sqrt{n}}$$

Hypothesis Testing

Test statistic follows t-distribution with $df = n - 1$ when testing means.

Examples

Example 1: Finding Critical Value

For 95% confidence interval with $n = 15$:

$$df = 15 - 1 = 14$$ $$\alpha = 0.05 \Rightarrow \alpha/2 = 0.025$$ $$t_{0.025, 14} = 2.145$$

Example 2: Confidence Interval

Sample: $n = 20$, $\bar{x} = 85$, $s = 12$

95% CI for $\mu$:

$$df = 19, \quad t_{0.025, 19} = 2.093$$ $$\text{CI} = 85 \pm 2.093 \times \frac{12}{\sqrt{20}} = 85 \pm 2.093 \times 2.683 = 85 \pm 5.62$$ $$\text{CI} = (79.38, 90.62)$$

Example 3: T-Test

Testing $H_0: \mu = 100$ vs $H_1: \mu \neq 100$

Sample: $n = 25$, $\bar{x} = 95$, $s = 15$

$$t = \frac{95 - 100}{15/\sqrt{25}} = \frac{-5}{3} = -1.67$$ $$df = 24, \quad \text{Critical value: } t_{0.025, 24} = 2.064$$ $$\lvert -1.67 \rvert < 2.064 \Rightarrow \text{fail to reject } H_0$$

When to Use Z vs T

Situation	Distribution
$\sigma$ known	Z
$\sigma$ unknown, $n \geq 30$	Z or T (similar)
$\sigma$ unknown, $n < 30$	T
Proportions	Z

Assumptions for T-Tests

1. Random sampling
2. Independence of observations
3. Population is approximately normal (or $n \geq 30$)