Interpreting and justifying claims with a confidence interval for a mean

What the exam asks

The exam tests four specific tasks:

• Interpret a confidence interval in context by referring to the population mean $\mu$ or population mean difference $\mu_d$ (Skill 4.F).
• Interpret a confidence level as a long-run capture rate across repeated random samples of the same size from the same population (Skill 4.F).
• Justify a claim about $\mu$ or $\mu_d$ by checking whether the interval of plausible values supports or contradicts the claim (Skill 4.G).
• Describe how changing the confidence level or sample size alters the critical value $t^*$, the margin of error, and the interval width (Skill 2.D).

Core ideas

Interpreting the confidence interval vs. the confidence level

Students often confuse interpreting the interval (the specific pair of numbers calculated from your one sample) with interpreting the confidence level (the reliability of the method across many samples).

Justifying claims with a confidence interval

Every value inside a confidence interval is considered a plausible value for the parameter $\mu$ or $\mu_d$. Values outside the interval are implausible.

• Claims about a single value ($\mu = k$ or $\mu_d = 0$):
  • If $k$ is inside the interval, $$ is a plausible value; we have convincing statistical evidence against the claim that .