AP Statistics Unit 7: Confidence Intervals for Quantitative Means (One-Sample and Two-Sample)

Population mean (μ)

The true (unknown) average of a population; the parameter a confidence interval for a mean aims to estimate.

Sample mean (x̄)

The average computed from sample data; used as the point estimate for the population mean μ.

Population standard deviation (σ)

The true (usually unknown) standard deviation of a population; if known, z procedures for means can be used.

Sample standard deviation (s)

The standard deviation computed from sample data; used to estimate σ when σ is unknown.

z statistic (for a mean)

Standardized statistic z = (x̄ − μ)/(σ/√n), which follows the standard normal distribution when σ is known and conditions are met.

t-distribution

A family of symmetric, bell-shaped distributions (centered at 0) used for inference about a mean when σ is unknown and replaced by s; has heavier tails than the normal.

t statistic

Standardized statistic t = (x̄ − μ)/(s/√n), used when σ is unknown; follows a t-distribution if conditions are met.

Degrees of freedom (df)

A value that determines the exact shape of a t-distribution; reflects how much independent information is available to estimate variability.

One-sample degrees of freedom (df = n − 1)

For one-sample t procedures, df equals n − 1 because estimating s uses the sample mean x̄, costing one degree of freedom.

Heavier tails

A feature of t-distributions (especially with small df) meaning more probability in the extremes than the standard normal, leading to larger critical values and wider intervals.

t approaches normal as df increases

As sample size (and df) get larger, the t-distribution becomes closer to the standard normal, so t critical values get closer to z critical values.

Critical value (t*)

The t value (based on df and confidence level C) that captures the middle C of the t-distribution; used to build a t confidence interval.

Confidence interval for a population mean

A range of plausible values for μ constructed from a point estimate (typically x̄) plus/minus a margin of error.

Point estimate

A single best guess for a parameter based on sample data (e.g., x̄ estimates μ; x̄1 − x̄2 estimates μ1 − μ2).

Standard error (SE) of x̄

The estimated standard deviation of the sampling distribution of x̄ when σ is unknown: SE = s/√n.

One-sample t confidence interval formula

x̄ ± t(s/√n), where t comes from the t-distribution with df = n − 1.

Margin of error (ME)

The amount added/subtracted from the point estimate in a confidence interval; for a t interval, ME = t*(s/√n).

Correct meaning of a 95% confidence interval

In the long run, about 95% of intervals made by the method will capture μ; a specific computed interval either contains μ or it does not.

Random condition (for t intervals)

Requirement that data come from a random sample or randomized experiment to justify inference.

Normal/Approximately Normal condition

Requirement that the sampling distribution of the mean is approximately normal (population roughly normal, or large n via CLT; for small n, sample should be roughly symmetric with no strong outliers).

Independence condition

Requirement that observations are independent; commonly supported in sampling without replacement by the 10% condition.

10% condition

Check for independence when sampling without replacement: n ≤ 0.10N (sample size is at most 10% of the population).

Two-sample t confidence interval (independent groups)

Interval for μ1 − μ2: (x̄1 − x̄2) ± t*√(s1²/n1 + s2²/n2), used when comparing two independent groups with unknown σ1 and σ2.

Standard error for difference of two means

SE = √(s1²/n1 + s2²/n2), combining variability from both samples when estimating μ1 − μ2.

Paired vs. independent samples

Paired data are matched (same individuals before/after or matched pairs) and should use a one-sample t interval on the differences; independent groups use the two-sample t interval for μ1 − μ2.