Statistics · CED Unit 7: Inference for Quantitative Data: Means · 14 min read · Updated 2026-05-11
Inference for a Population Mean — AP Statistics
AP Statistics · CED Unit 7: Inference for Quantitative Data: Means · 14 min read
1. Overview of Inference for a Population Mean★★☆☆☆⏱ 2 min
Inference for a population mean uses quantitative sample data to draw evidence-based conclusions about the unknown true population mean $\mu$. This topic makes up roughly 4-5% of the total AP Statistics exam score, appearing regularly in both multiple-choice and free-response questions.
2. The t-Distribution and Conditions for Inference★★☆☆☆⏱ 3 min
When we do not know the true population standard deviation $\sigma$, we estimate it with the sample standard deviation $s$, giving us the *standard error* of the sample mean: $s/\sqrt{n}$. The ratio $\frac{\bar{x} - \mu}{s/\sqrt{n}}$ follows a t-distribution, not a normal distribution.
The t-distribution is symmetric, bell-shaped, and centered at 0, similar to the z-distribution, but has fatter tails to account for the extra variability from estimating $\sigma$ with $s$. The shape depends only on degrees of freedom, where for one-sample inference, $df = n-1$. As $df$ (and sample size) increases, the t-distribution approaches the z-distribution.
**Random**: Data comes from a random sample or randomized experiment to ensure unbiasedness.
**Independence**: Individual observations are independent. For sampling without replacement, the 10% condition requires $n < 0.1N$.
**Normal/Large Sample**: The sampling distribution of $\bar{x}$ is approximately normal if $n \geq 30$ (Central Limit Theorem) or, for smaller $n$, the sample has no strong skewness or outliers.
3. One-Sample t-Interval for a Population Mean★★★☆☆⏱ 3 min
A one-sample t-interval estimates the unknown population mean $\mu$ from sample data, following the general confidence interval structure:
For a population mean, the point estimate is $\bar{x}$, the critical value is $t^*_{df}$ (from a t-table or calculator, matching your confidence level and $df = n-1$), and the standard error is $s/\sqrt{n}$. The full formula is:
\bar{x} \pm t^*_{df} \frac{s}{\sqrt{n}}
The correct interpretation of a C% confidence interval is: *We are C% confident that the interval from [lower bound] to [upper bound] captures the true population mean [in context of the problem]*. The confidence level describes the long-run behavior of the method: if we repeated sampling many times, C% of all intervals constructed this way would capture the true mean.
4. One-Sample t-Test for a Population Mean★★★☆☆⏱ 3 min
A one-sample t-test is used to test a claim about the value of a population mean $\mu$. The null hypothesis is always $H_0: \mu = \mu_0$, where $\mu_0$ is the hypothesized value from the claim. The alternative hypothesis is two-sided ($H_a: \mu \neq \mu_0$), left-tailed ($H_a: \mu < \mu_0$), or right-tailed ($H_a: \mu > \mu_0$), depending on the research question.
The test statistic for a one-sample t-test is:
t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}
The p-value is the probability of observing a t-statistic as extreme or more extreme than the one calculated, assuming $H_0$ is true. We compare the p-value to the significance level $\alpha$ (usually 0.05): if $p < \alpha$, we reject $H_0$; otherwise, we fail to reject $H_0$.
5. Paired t-Procedures for Dependent Samples★★★★☆⏱ 3 min
Paired data occurs when we have two dependent measurements (e.g., before/after treatment on the same subject, matched pairs of similar subjects). Because the two measurements are not independent, we cannot use two-sample t-procedures. Instead, we calculate the difference $d_i$ for each pair, then conduct one-sample inference on the true mean difference $\mu_d$.
All rules for one-sample t-intervals and t-tests apply directly to paired data: $df = n - 1$ where $n$ is the number of pairs, and we use the mean difference $\bar{d}$ and standard deviation of differences $s_d$ in all calculations.
Common Pitfalls
Why: Students confuse inference for means (almost always use t) with inference for proportions (always use z).
Why: Students memorize the $n \geq 30$ rule and forget that normality can still be assumed for small samples with roughly symmetric data.
Why: Students confuse the location of the true mean with the behavior of the sampling method.
Why: Students think a large p-value proves the null hypothesis is true.
Why: Students see two groups of data and automatically jump to a two-sample test, without noticing the pairing.
Why: Students mix up sample size and degrees of freedom, leading to incorrect critical values and p-values.