Significance Tests for Means

One-Sample t-Test for $\mu$

Hypotheses:

• $H_0: \mu = \mu_0$
• $H_a: \mu \neq \mu_0$ (or $>$ or $<$)

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

Degrees of freedom: $df = n - 1$

Conditions:

1. Random sample or randomized experiment
2. 10% Condition: $n < 0.10N$
3. Normal/Large Sample: Population is Normal, or $n \geq 30$ (CLT), or graph shows no strong skewness/outliers

Two-Sample t-Test for $\mu_1 - \mu_2$

Hypotheses:

• $H_0: \mu_1 = \mu_2$ (equivalently, $\mu_1 - \mu_2 = 0$)
• $H_a: \mu_1 \neq \mu_2$ (or $>$ or $<$)

Test Statistic: $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Degrees of freedom: Use the calculator's value (Welch's approximation), or conservatively use $df = \min(n_1 - 1, n_2 - 1)$.

Conditions (check for BOTH samples):

1. Random: Both samples are independently random
2. 10%: Both $n_1 < 0.10N_1$ and $n_2 < 0.10N_2$
3. Normal/Large Sample: Both populations Normal, or both $n \geq 30$

Two-Sample t-Interval for $\mu_1 - \mu_2$

$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Paired t-Test

For matched pairs or before/after data:

1. Compute differences: $d_i = x_{1i} - x_{2i}$
2. Apply one-sample t-test to the differences

$t = \frac{\bar{d} - 0}{s_d / \sqrt{n}}$

with $df = n - 1$ (number of pairs minus 1)

Important Notes

• Do not pool standard deviations for the two-sample t-test (pooled t-test is rarely used)
• The two samples must be independent of each other
• If subjects are matched or measured twice, use the paired t-test

Summary: Which Test?

| Situation | Test | |-----------|------| | One sample, unknown $\sigma$ | One-sample t-test | | Two independent samples | Two-sample t-test | | Matched pairs | Paired t-test | | One proportion | One-sample z-test | | Two proportions | Two-sample z-test |

AP Tip: On the AP exam, always use the t-test for means (not z-test). Show all four steps: State, Plan, Do, Conclude.