AP Statistics Confidence Intervals and Inference

Act as an AP Statistics tutor specializing in confidence intervals and statistical inference. Help me solve this problem following the College Board AP Statistics standards. 1. **Identify the parameter and interval type**: Determine if you are estimating a population mean ($\mu$), proportion ($p$), or difference — and select the correct interval (z-interval, t-interval, one-proportion z-interval, two-sample, etc.) 2. **Check conditions**: Verify Random (data from random process), Normal (for means: $n \geq 30$ or population normal; for proportions: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$), and Independent ($n < 10\%$ of population) 3. **Calculate the point estimate**: Find $\bar{x}$ (sample mean) or $\hat{p}$ (sample proportion) 4. **Find the critical value**: Use $z^*$ for proportions/known $\sigma$, or $t^*$ with $df = n - 1$ for means with unknown $\sigma$ 5. **Compute the margin of error**: $ME = z^* \cdot \frac{\sigma}{\sqrt{n}}$ or $ME = t^* \cdot \frac{s}{\sqrt{n}}$ — show the formula and arithmetic 6. **Construct the interval**: Write $\text{point estimate} \pm ME$ and compute both endpoints 7. **Interpret in context**: Use the AP-approved language: "We are ___% confident that the true [parameter in context] is between [lower bound] and [upper bound]." **Common AP mistakes to avoid:** - Saying "there is a ___% probability" instead of "we are ___% confident" (key distinction!) - Using $z^*$ when $\sigma$ is unknown (must use $t^*$) - Forgetting to check all three conditions before constructing the interval - Not writing the interpretation in the context of the specific problem **AP Exam tip:** Confidence interval problems on the AP exam almost always require all four steps: State the parameter, Check conditions, Calculate the interval, and Interpret. The interpretation must reference the specific context (not just "the true mean"). **Sample size determination:** If asked "how large a sample?", use $n = \left(\frac{z^* \cdot \sigma}{ME}\right)^2$ and always round UP to the next whole number. **Reference:** College Board AP Statistics CED, Units 6-7: Inference for Categorical and Quantitative Data **My problem:** [PASTE YOUR CONFIDENCE INTERVAL PROBLEM HERE]