4.4 Setting Up a Test for a Population Mean

Hypotheses for a mean

State a null hypothesis H₀: μ = value and an alternative hypothesis Hₐ: μ < , > , or ≠ value, based on the research question. Hypotheses are always about the population parameter μ, never the sample mean x̄. Choose the direction of Hₐ before looking at the data.

Checking conditions

Before running a t-test for a mean, verify: (1) Random — data from a random sample or randomized experiment; (2) Independence — 10% condition if sampling without replacement; (3) Normality — population is normal, or sample size is large enough (n ≥ 30, or check for strong skew/outliers with graphs if n is small) to use the Central Limit Theorem.

Test statistic and df

The test statistic is t = (x̄ − μ₀) / (s/√n), following a t-distribution with n − 1 degrees of freedom since σ is unknown and estimated by sample standard deviation s. Larger df make the t-distribution closer to normal.

Key terms

Null hypothesis (mean test)

H₀: μ = hypothesized value; states there is no effect or difference in the population mean.

Alternative hypothesis (mean test)

Hₐ: μ <, >, or ≠ hypothesized value; states the direction or existence of a difference.

Conditions for a t-test for μ

Random sample, Independence (10% condition if sampling without replacement), and Normality (population normal or n large enough / no strong skew/outliers).

t test statistic formula

t = (x̄ − μ₀) / (s/√n), using sample standard deviation s in place of unknown σ.

Degrees of freedom (one-sample t-test)

df = n − 1, where n is the sample size.

Try an AP-style question

A company claims its batteries last 12 hours on average. A tester suspects they last LESS. Which hypotheses fit?

1. H₀: μ = 12; Hₐ: μ > 12
2. H₀: μ < 12; Hₐ: μ = 12
3. H₀: μ = 12; Hₐ: μ < 12
4. H₀: x̄ = 12; Hₐ: x̄ < 12