Where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean
• $s$ = sample standard deviation
• $n$ = sample size
• Degrees of freedom: $df = n - 1$

Conditions:

1. Random sample from population
2. Independent observations (or $n \leq 0.1N$)
3. Approximately normal: Data roughly normal OR $n \geq 30$ (CLT)

Worked Example: A coffee shop claims their average cup is 16 oz. A random sample of 25 cups has mean $\bar{x} = 15.2$ oz, SD $s = 1.8$ oz. Test at $\alpha = 0.05$.

• $t = \frac{15.2 - 16}{1.8/\sqrt{25}} = \frac{-0.8}{0.36} = -2.22$
• $df = 24$; two-tailed critical value $t^* = 2.064$
• Since $|-2.22| > 2.064$, reject $H_0$
• Conclusion: Average cup is significantly below 16 oz

Two-Sample t-Test (Welch's Test)

When to use: Comparing two populations; testing $\mu_1 - \mu_2 = 0$

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 (Welch): Complex formula (calculators handle this)

Conditions: Both samples random, independent, approximately normal

Use: When sample sizes unequal or variances appear different

Common Mistakes

❌ Using z-test instead of t-test for means (unknown $\sigma$) ❌ Using pooled t-test when variances unequal (use Welch) ❌ Confusing $SE = s/\sqrt{n}$ with $s$ (the standard deviation) ❌ Assuming normality without checking

Decision Rule

• If $|t| > t^*_{df, \alpha/2}$, reject $H_0$
• If p-value $< \alpha$, reject $H_0$

AP Exam Tip

Always specify degrees of freedom. Name the test clearly: "one-sample t-test" or "two-sample t-test." Mention whether conditions are met.