t-distributions and the one-sample t-interval for a mean or mean difference

What the exam asks

• Identify a one-sample $t$-interval for a population mean ($\mu$) or for a population mean difference ($\mu_d$) from matched pairs, and define the parameter in context.
• Describe $t$-distributions and explain why $t^*$ replaces $z^*$ when $\sigma$ is unknown.
• Justify the randomization, 10%, and sample data (normality) conditions using numbers and graphs from the prompt.
• Calculate the standard error, critical value $t^*$ with $df = n - 1$, margin of error, and confidence interval.

Core ideas

Why we use the $t$-distribution

The sampling distribution of $\bar{x}$ has standard deviation $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$. Because we rarely know the population standard deviation , we estimate it with the sample standard deviation , giving the of :

Replacing the fixed parameter $\sigma$ with the random variable $s$ introduces extra sample-to-sample variability. Consequently, the standardized statistic $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ does not follow a standard normal () distribution; it follows a .

Key properties of $t$-distributions:

• Symmetric, unimodal, and bell-shaped, centered at $0$ (like the $z$-distribution).
• More area in the tails (heavier tails and greater spread) than the standard normal distribution, so $t^* > z^*$ for any given confidence level.
• As the sample size and degrees of freedom increase, becomes a more reliable estimate of , and the .

One-sample vs. matched pairs $t$-intervals

When data come from a single sample or from matched pairs (either two measurements on each unit or pairs of similar units with treatments randomly assigned within each pair), we collapse paired data into a single list of differences ($d = \text{first} - \text{second}$) and apply a one-sample procedure to those differences.