Selecting Procedures for Means

First determine how many groups or samples you have. If you are comparing a sample mean to a specific fixed value (e.g., 'Is the average battery life equal to 10 hours?'), you need a one-sample procedure. If you are comparing two groups to each other (e.g., 'Do students in School A score higher than students in School B?'), you need a two-sample procedure. This single question immediately narrows your options significantly.

When you have two sets of measurements, determine whether the observations are naturally paired. Data is paired when each measurement in Group 1 is logically linked to exactly one measurement in Group 2 — for example, measuring the same person's blood pressure before and after a treatment, or comparing twins raised in different environments. If data is paired, compute the differences d = x₁ - x₂ for each pair and apply a one-sample t-procedure to those differences. Treating paired data as independent two-sample data wastes information and reduces the power of your test.

Ask whether the population standard deviation σ is known. In the extremely rare case that σ is known (sometimes assumed in textbook problems to introduce the z-procedure), you use a z-test or z-interval: the test statistic is z = (x̄ − μ₀) / (σ / √n). In virtually all real-world settings, σ is unknown and must be estimated by the sample standard deviation s. In this case, use a t-procedure with t = (x̄ − μ₀) / (s / √n), which follows a t-distribution with n − 1 degrees of freedom.

Before proceeding with any procedure, check three conditions. (1) Random: The data must come from a random sample or a randomized experiment. This ensures the results generalize to the population. (2) Independence / 10% condition: Individual observations must be independent. When sampling without replacement, the sample size n must be at most 10% of the population size N, so that n ≤ 0.10·N. (3) Normality: The sampling distribution of x̄ must be approximately normal. This is satisfied if the population is normal, or by the Central Limit Theorem when n ≥ 30. For smaller samples, examine the data for severe skewness or outliers using a dotplot or boxplot.

When comparing two independent groups and σ is unknown (the standard case), use the two-sample t-test. The test statistic is t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂). The degrees of freedom are computed using the Welch-Satterthwaite approximation (a complex formula), but calculators and software handle this automatically. Do NOT assume equal variances (pooled t) unless you have a strong, verified reason to do so — the unpooled Welch procedure is safer and is the standard in AP Statistics and most introductory courses.

• \bar{x} — Sample mean (same as data): The average of all values in the sample, computed as the sum of observations divided by n.
• \mu_0 — Hypothesized population mean (same as data): The specific value of the population mean stated in the null hypothesis H₀.
• s — Sample standard deviation (same as data): The standard deviation calculated from the sample, used to estimate the unknown population standard deviation σ.
• n — Sample size (count): The number of observations in the sample. Larger n reduces the standard error and increases test power.
• s / \sqrt{n} — Standard error of the mean (same as data): Estimates how much the sample mean x̄ varies from sample to sample due to random chance.

Problem. A school nutritionist claims the average number of calories in the cafeteria's lunch is μ = 650 calories. A random sample of 20 lunches gives a mean of x̄ = 682 calories with a standard deviation of s = 48 calories. At the α = 0.05 significance level, is there evidence that the true mean calorie count differs from 650? State which procedure you would use and why, then carry out the test.

Answer. Use the one-sample t-test (σ unknown, one group, compared to fixed value). Reject H₀; p ≈ 0.0078 < 0.05.

• One-Sample t-Test — The one-sample t-test is the most common procedure identified in the selection process when comparing a sample mean to a fixed value with unknown σ.
• Two-Sample t-Test — Selected when comparing means from two independent groups; understanding when to use this versus the paired t-test is central to procedure selection.
• Paired t-Test — Recognizing paired data structures is a key decision point in choosing the correct inference procedure for means.
• Conditions for Inference for Means — Before any procedure can be applied, the three conditions (Random, Independence, Normality) must be verified — these are central to valid inference.
• The t-Distribution — All t-procedures rely on the t-distribution for calculating p-values and critical values; understanding its shape and degrees of freedom is essential.