Inference for Means: t-Based Significance Testing (AP Statistics Unit 7)

Significance test for a population mean

A procedure that uses sample data to judge whether there is convincing evidence about a claim regarding a population mean (μ) compared to a hypothesized value.

Population mean (μ)

The true average value of a quantitative variable for the entire population; the parameter targeted in a one-sample mean test.

Hypothesized mean (μ0)

The benchmark value for the mean stated in the null hypothesis (e.g., advertised average or historical mean).

Null hypothesis (H0)

A statement about a population parameter representing “no effect” or “no difference,” typically using equality (e.g., μ = μ0).

Alternative hypothesis (Ha)

The claim the test seeks evidence for (e.g., μ ≠ μ0, μ > μ0, or μ < μ0); its direction must match the research question.

Two-sided (two-tailed) test

A test with Ha: parameter ≠ hypothesized value; looks for differences in either direction.

Right-tailed test

A test with Ha: parameter > hypothesized value; evidence is in the upper tail of the sampling distribution.

Left-tailed test

A test with Ha: parameter < hypothesized value; evidence is in the lower tail of the sampling distribution.

Student’s t distribution

A family of symmetric distributions with heavier tails than the normal distribution, used for mean inference when the population standard deviation is unknown.

Degrees of freedom (df)

A parameter that determines the exact t distribution used; for a one-sample t procedure, df = n − 1.

t test statistic (one-sample)

t = (x̄ − μ0) / (s/√n); measures how many standard errors the sample mean is from the hypothesized mean.

Standard error of the mean (s/√n)

The typical sample-to-sample variability of the sample mean x̄, estimated using the sample standard deviation s.

p-value

Assuming H0 is true, the probability of observing a test statistic as extreme or more extreme than the one obtained, in the direction(s) of Ha.

Significance level (α)

The cutoff probability for deciding whether results are statistically significant (commonly 0.05); compared to the p-value.

Reject H0

Decision made when p ≤ α; conclude the data provide convincing evidence for Ha (in context).

Fail to reject H0

Decision made when p > α; conclude the data do not provide convincing evidence for Ha (not the same as proving H0 true).

Random condition

Requirement that data come from a random sample or random assignment; supports valid inference (and affects generalization vs causation).

Independence condition

Requirement that observations are independent; for sampling without replacement, often checked using the 10% condition.

10% condition

A guideline for independence when sampling without replacement: the sample size n should be ≤ 0.1N (population size).

Normal/large sample condition

For t procedures, the population distribution is approximately normal or the sample size is large enough for x̄ (or differences) to be approximately normal; outliers/skew can still be a problem.

Two-sample (independent) significance test for μ1 − μ2

A test comparing two population means from independent groups; targets the parameter μ1 − μ2.

Two-sample t test statistic (unpooled)

t = ((x̄1 − x̄2) − Δ0) / √(s1²/n1 + s2²/n2); compares the observed difference in sample means to the null difference.

Welch’s degrees of freedom (unpooled df)

An approximate df used for the standard (unpooled) two-sample t procedure; often provided by technology (a conservative hand option is min(n1−1, n2−1)).

Matched pairs design

A design where each observation in one condition is paired with a related observation in the other (e.g., before/after or matched subjects), reducing variability by focusing on within-pair change.

Mean difference (μd)

The population mean of the paired differences d; the parameter used in matched pairs t procedures after computing d for each pair.