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Introductory Statistics Unit 8: Inference for means

When the data are measurements, you estimate or test a population mean with t-procedures, because the population SD is usually unknown.

Covers: t-procedures for one mean, two independent means, matched pairs

Study guide

Inference for means uses the t-distribution because the population standard deviation σ is almost never known and is estimated by the sample standard deviation s. The t-distributions are bell-shaped like the normal but have heavier tails, and they depend on degrees of freedom: for one sample, df = n − 1. As df increases, t approaches the standard normal.

A one-sample t interval is x̄ ± t*·s/√n, and a one-sample t test uses t = (x̄ − μ₀)/(s/√n). Conditions: random sample, the 10% condition, and Normal/large sample: the population is normal, or n ≥ 30, or for small samples a graph of the data shows no strong skew or outliers.

Two independent samples compare two population means with (x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂), and the test statistic t = (x̄₁ − x̄₂)/√(s₁²/n₁ + s₂²/n₂). Use technology for the degrees of freedom, or the smaller of n₁ − 1 and n₂ − 1 as a conservative choice. Check conditions for each group.

Matched pairs data come from the same individuals measured twice (before and after) or from paired subjects. Don't use two-sample methods: compute the difference for each pair and run a one-sample t procedure on the differences, with μ_d as the parameter.

Interpret intervals and tests in context, with units, and connect them: a 95% interval that excludes μ₀ corresponds to rejecting H₀ at α = 0.05 in a two-sided test.

Statistical significance doesn't guarantee practical importance, because very large samples can make tiny differences significant. An effect size such as Cohen's d = (x̄₁ − x̄₂)/s_pooled measures a difference in standard-deviation units: about 0.2 is small, 0.5 medium and 0.8 large. Define the parameter in words with context, for example "μ = the true mean commute time, in minutes, for all students at the school."

A one-sided test uses the area in the tail that points in the direction of Hₐ. A two-sided test doubles the area of one tail. Greater variability (a larger s) widens confidence intervals, shrinks test statistics, and lowers both power and effect size.

Key ideas

• Use t instead of z when you estimate σ with the sample SD s.
• t-distributions have heavier tails than the normal; they approach normal as df grows.
• Conditions: random, 10%, and normal population or n ≥ 30 (or no strong skew/outliers for small samples).
• Matched pairs: take the differences and run a one-sample t on them.
• Two-sample t compares means of two independent groups.

Formulas and rules

Common mistakes

• Using a two-sample test on paired data.
• Using z* with a sample SD.

Words to know

One-sample t interval

x̄ ± t*·s/√n with df = n − 1

One-sample t test statistic

t = (x̄ − μ₀) / (s/√n)

t-distribution

Used for means when σ is unknown and estimated by s.

Matched pairs

Analyze the differences with a one-sample t procedure.

Normal/large sample condition for t

Population is normal, or n ≥ 30, or a small sample shows no strong skew or outliers.

Degrees of freedom (t)

As df increases, t approaches the standard normal.

Two-sample t statistic

t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Standard error of the mean

s/√n

t* critical value

From the t-distribution with n − 1 df for the chosen confidence level.

Two-sample t interval

(x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂)

Matched pairs parameter (μ_d)

The mean of the differences.

Robust procedure

Still accurate when conditions are mildly violated; t procedures are robust for large n.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.