Confidence Interval Formulas

Why This Matters

Confidence intervals are the backbone of statistical inference on the AP Statistics exam. They show up in Units 6, 7, and 9, and you'll encounter them in both multiple-choice questions and FRQs. The College Board wants you to understand that we use intervals (not single values) to estimate population parameters because sample statistics vary from sample to sample. Every confidence interval you construct reflects this fundamental truth: you're acknowledging uncertainty while still making useful claims about populations.

Here's what you're really being tested on: knowing which interval procedure fits which situation, verifying the conditions that make each formula valid, and interpreting your results correctly. The formulas themselves follow a consistent structure: point estimate ± (critical value)(standard error). But the details change depending on whether you're estimating proportions vs. means, one sample vs. two samples, or categorical vs. quantitative relationships. Don't just memorize formulas; know what type of data and research question each one addresses.

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One-Sample Intervals for Proportions

When you have categorical data from a single sample and want to estimate the true population proportion, you'll use the one-sample z-interval. The sampling distribution of $\hat{p}$ is approximately normal when sample sizes are large enough, which is why we can use the standard normal (z) distribution here.

One-Sample Z-Interval for a Proportion

• Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\hat{p}$ is the sample proportion and $z^*$ is the critical value (e.g., $z^* = 1.96$ for 95% confidence)
• Success-failure condition: both $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$ must be satisfied. This ensures the sampling distribution is approximately normal so that using $z^*$ is valid.
• Independence conditions: data must come from a random sample or randomized experiment, and the 10% condition ($n \leq 0.10N$) applies when sampling without replacement. Without independence, the standard error formula breaks down.

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One-Sample Intervals for Means

When estimating a population mean from quantitative data, the choice between z and t depends on whether you know the population standard deviation. In practice, you almost never know $\sigma$, so the t-interval dominates AP Statistics.

Z-Interval for a Mean (Known $\sigma$)

• Formula: $\bar{x} \pm z^* \left(\frac{\sigma}{\sqrt{n}}\right)$ using the known population standard deviation $\sigma$ in the standard error
• Rarely used in practice because knowing $\sigma$ while not knowing $\mu$ is an unusual situation. This formula appears mainly in theoretical or textbook problems.
• Normality required: either the population is normally distributed, or $n \geq 30$ so the Central Limit Theorem kicks in

T-Interval for a Mean (Unknown $\sigma$)

• Formula: $\bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right)$ where $s$ replaces $\sigma$, with degrees of freedom $df = n - 1$
• The t-distribution is wider than the z-distribution, especially for small samples. Those heavier tails account for the extra uncertainty that comes from estimating $\sigma$ with $s$.
• Conditions: random sample, independence (10% condition), and population approximately normal OR large sample size ($n \geq 30$)

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Two-Sample Intervals for Comparing Groups

Comparing two populations is where inference gets more interesting. The key question: are the samples independent (two separate groups) or paired (same subjects measured twice)?

Two-Sample T-Interval for Difference of Means

• Formula: $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ where the standard error combines variability from both samples
• Degrees of freedom: use your calculator's "2-SampTInt" function, which applies the Welch approximation. Don't try to compute df by hand on the AP exam.
• Conditions: two independent random samples, 10% condition for each group, and both populations approximately normal OR both sample sizes large

Two-Sample Z-Interval for Difference of Proportions

• Formula: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$ Note that you use each sample's $\hat{p}$ separately. There's no pooling for confidence intervals (pooling only happens in hypothesis tests for proportions).
• Interpretation: if the interval contains zero, you don't have convincing evidence of a difference. The sign of the values tells you which group has the larger proportion.
• Success-failure condition: check all four values: $n_1\hat{p}_1$, $n_1(1-\hat{p}_1)$, $n_2\hat{p}_2$, and $n_2(1-\hat{p}_2)$. Each must be $\geq 10$.

Paired T-Interval for Mean Difference

• Formula: $\bar{d} \pm t^* \left(\frac{s_d}{\sqrt{n}}\right)$ where $\bar{d}$ is the mean of the differences and $s_d$ is the standard deviation of the differences
• When to use: matched pairs designs, before-and-after studies, or any situation where each observation in one sample is linked to a specific observation in the other
• The core idea: you're reducing a two-sample problem to a one-sample problem by computing the differences first, then analyzing those differences. Here $df = n - 1$ where $n$ is the number of pairs.

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Inference for Regression Slopes

Unit 9 extends confidence intervals to linear regression. Here you're estimating the true population slope $\beta$ based on your sample slope $b$.

T-Interval for the Slope of a Regression Line

• Formula: $b \pm t^* \cdot SE_b$ where $SE_b = \frac{s}{\sqrt{\sum(x_i - \bar{x})^2}}$ and $s$ is the residual standard deviation. In practice, your calculator or computer output provides $SE_b$ directly.
• Degrees of freedom: $df = n - 2$. You lose two degrees of freedom because you're estimating both the slope and the intercept.
• Conditions (LINE):
  • Linear relationship (check the residual plot for no curved pattern)
  • Independence (random sample, 10% condition)
  • Normal residuals (check with a histogram or Normal probability plot of residuals)
  • Equal variance (residuals show constant spread across all x-values, no "fan" shape)

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Advanced Intervals (Beyond Core AP Content)

These formulas occasionally appear in enrichment contexts but are not central to the AP Statistics exam. Know they exist, but prioritize the intervals above.

Confidence Interval for Population Variance

• Formula: $\left(\frac{(n-1)s^2}{\chi^2_{\alpha/2}},\ \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}\right)$ using the chi-squared distribution, which is right-skewed
• Not symmetric: unlike z and t intervals, this interval is asymmetric around the point estimate
• Strong normality assumption: the population must be normally distributed. This procedure is sensitive to departures from normality.

Confidence Interval for Ratio of Two Variances

• Formula: $\left(\frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{\alpha/2}},\ \frac{s_1^2}{s_2^2} \cdot F_{\alpha/2}\right)$ using the F-distribution with $df_1 = n_1 - 1$ and $df_2 = n_2 - 1$
• Application: testing whether two populations have equal variances before running a pooled two-sample t-test
• Requires normality in both populations. Rarely tested on AP Statistics but useful for understanding ANOVA assumptions.

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Quick Reference Table

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Self-Check Questions

1. What conditions must you verify before constructing a one-sample z-interval for a proportion, and why does each condition matter?

2. Compare the t-interval for a single mean and the paired t-interval: what do they have in common, and when would you choose one over the other?

3. If a confidence interval for $p_1 - p_2$ is $(0.03, 0.15)$, what can you conclude about the relationship between the two population proportions?

4. An FRQ gives you regression output including $b = 2.4$ and $SE_b = 0.6$ with $n = 22$. What critical value would you use for a 95% confidence interval, and what are the degrees of freedom?

5. A researcher wants to determine whether a new teaching method improves test scores. Students are tested before and after the intervention. Which confidence interval procedure is appropriate, and why would using a two-sample t-interval be incorrect here?