One-sample t-test for a population mean or mean difference

What the exam asks

On both sections of the digital exam, you must be ready to:

• Identify the procedure by its exact name and define the parameter ($\mu$ or $\mu_d$) in context.
• State the null hypothesis $H_0$ and a one- or two-sided alternative hypothesis $H_a$.
• Justify the three conditions: randomization, the $10\%$ condition (when sampling without replacement), and the sample data (normality) condition.
• Calculate the test statistic $t$ and the p-value using technology or Table B.
• Interpret the p-value in context and justify a conclusion by comparing the p-value to the significance level $\alpha$.

Core ideas

1. Identifying the procedure and parameter

Because the population standard deviation $\sigma$ is almost never known when testing a mean, you estimate it using the sample standard deviation $s$. This requires a $t$-distribution with $df = n - 1$.

2. Hypotheses

Hypotheses are always written about population parameters ($\mu$ or $\mu_d$), never sample statistics ($\bar{x}$ or $\bar{x}_d$).

• Null hypothesis: $H_0: \mu = \mu_0$ (for matched pairs, usually $H_0: \mu_d = 0$).

3. Conditions for inference

Verify every condition in context using numbers and facts from the prompt:

1. Randomization: The data come from a random sample from the population of interest, or treatments were randomly assigned to pairs in an experiment.
2. $10\%$ Condition: When sampling without replacement, the sample size $n$ (or number of pairs) must be at most $10\%$ of the population size ($n \le 0.10N$). Skip this condition in a randomized experiment that does not sample from a finite population.
3. Sample Data (Normality) Condition: Check one of the following:
   • The population distribution (or population of differences) is stated to be approximately normal.