T-Test for Means

Overview

The t-test is used to test hypotheses about population means when the population standard deviation is unknown and must be estimated from sample data.

Types of T-Tests

Type	Purpose
One-sample	Compare sample mean to hypothesized value
Independent two-sample	Compare means of two independent groups
Paired (dependent)	Compare means of matched pairs or before/after

One-Sample T-Test

Test Statistic

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

Degrees of Freedom

$$df = n - 1$$

Independent Two-Sample T-Test

Equal Variances Assumed (Pooled)

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$$

Pooled variance:

$$s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}$$

Degrees of freedom:

$$df = n_1 + n_2 - 2$$

Unequal Variances (Welch's T-Test)

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$

Degrees of freedom (Welch-Satterthwaite):

$$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}$$

Paired T-Test

For differences $d = x_1 - x_2$:

$$t = \frac{\bar{d}}{s_d/\sqrt{n}}$$

Where:

• $\bar{d}$ = mean of differences
• $s_d$ = standard deviation of differences
• $df = n - 1$

Assumptions

1. Random sampling
2. Independence (except for paired data)
3. Normal population OR $n \geq 30$
4. For two-sample: homogeneity of variance (if using pooled)

Examples

Example 1: One-Sample T-Test

Test if mean differs from 75. Sample: $n = 16$, $\bar{x} = 78$, $s = 8$. $\alpha = 0.05$.

$$H_0: \mu = 75, \quad H_1: \mu \neq 75$$ $$t = \frac{78 - 75}{8/\sqrt{16}} = \frac{3}{2} = 1.5$$ $$df = 15, \quad t_{\text{crit}} = 2.131$$ $$\lvert 1.5 \rvert < 2.131 \Rightarrow \text{Fail to reject } H_0$$

Example 2: Independent Two-Sample (Pooled)

Group A: $n = 12$, $\bar{x} = 85$, $s = 6$

Group B: $n = 15$, $\bar{x} = 80$, $s = 8$

Test $H_0: \mu_1 = \mu_2$ at $\alpha = 0.05$.

$$s_p^2 = \frac{(11)(36) + (14)(64)}{25} = \frac{396 + 896}{25} = 51.68$$ $$t = \frac{85 - 80}{\sqrt{51.68\left(\frac{1}{12} + \frac{1}{15}\right)}} = \frac{5}{\sqrt{51.68(0.150)}} = \frac{5}{2.78} = 1.80$$ $$df = 25, \quad t_{\text{crit}} = 2.060$$ $$\lvert 1.80 \rvert < 2.060 \Rightarrow \text{Fail to reject } H_0$$

Example 3: Paired T-Test

Before/after measurements for 10 subjects:

Subject	Before	After	Difference ($d$)
Mean	-	-	$\bar{d} = 4.5$
SD	-	-	$s_d = 3.2$

Test $H_0: \mu_d = 0$ at $\alpha = 0.05$.

$$t = \frac{4.5}{3.2/\sqrt{10}} = \frac{4.5}{1.01} = 4.45$$ $$df = 9, \quad t_{\text{crit}} = 2.262$$ $$\lvert 4.45 \rvert > 2.262 \Rightarrow \text{Reject } H_0$$

Significant change from before to after.

Effect Size (Cohen's d)

$$d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}$$

$d$	Interpretation
0.2	Small
0.5	Medium
0.8	Large

Confidence Interval

For difference in means:

$$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, df} \times SE$$