Inference for Quantitative Data: Means

For a random sample, the sampling distribution of x̄ has mean μ and standard deviation σ/√n, so bigger samples give less variable means. It is normal whenever the population is normal, and approximately normal for large samples (n ≥ 30) even when the population isn't, thanks to the central limit theorem; a very skewed population may need a much bigger sample.

When σ is unknown you use s and a t-distribution: t-distributions are symmetric and bell-shaped with heavier tails than the standard normal, and which one you use depends on the degrees of freedom, df = n − 1. The one-sample t-interval is x̄ ± t* · s/√n, and for matched pairs you first take the difference within each pair and then build the same interval on those differences.

Interpret a t-interval as a range of plausible values for the population mean (or mean difference) in context, and use it to support or question a claim. Just as with proportions, a higher confidence level widens the interval and a larger sample narrows it.

A one-sample t-test checks a claim about a population mean, with H₀: μ = μ₀; for paired data you test the mean difference instead, with H₀: μd = 0. The conditions are a random sample or randomized experiment, the 10% condition when you sample without replacement, and either a population known to be roughly normal, a large sample (n ≥ 30) or a smaller sample with no strong skew or outliers (for paired data, check the differences).

The test statistic is t = (x̄ − μ₀) ÷ (s/√n) with df = n − 1, and you find the p-value from a t-table or technology. Compare the p-value with α and write a conclusion in context about the population mean or mean difference.

For two independent random samples, or a randomized experiment, the sampling distribution of x̄₁ − x̄₂ has mean μ₁ − μ₂ and standard deviation √(σ₁²/n₁ + σ₂²/n₂). It is normal if both populations are normal, and approximately normal when both samples are large (n₁ ≥ 30 and n₂ ≥ 30).

A two-sample t-interval estimates μ₁ − μ₂ with (x̄₁ − x̄₂) ± t* · √(s₁²/n₁ + s₂²/n₂). Check for two independent random samples or a randomized experiment, the 10% condition for each sample (not needed for a randomized experiment), and that both samples have n ≥ 30 or show no strong skew or outliers.

Interpret the interval as a range of plausible values for the difference between the two population means, in context. If 0 isn't in the interval, you have convincing evidence that the means differ; if 0 is inside it, "no difference" is still plausible.

A two-sample t-test checks whether two population means differ, starting from H₀: μ₁ = μ₂ (the same as μ₁ − μ₂ = 0) with a one-sided or two-sided alternative. It is for two independent groups; if the data come in pairs, use a paired t-test on the differences instead.

The test statistic is t = (x̄₁ − x̄₂ − 0) ÷ √(s₁²/n₁ + s₂²/n₂), and you find its p-value from a t-distribution, usually with technology. Compare the p-value with α and state a conclusion in context about the difference between the two population means.