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

Population mean (bc)

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

Sample mean ($\bar{x}$)

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

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 = \frac{\bar{x} - bc}{\frac{c3}{\sqrt{n}}}, which follows the standard normal distribution when c3 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 = \frac{\bar{x} - bc}{\frac{s}{\sqrt{n}}}, used when c3 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 $\bar{x}$, 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 bc constructed from a point estimate (typically $\bar{x}$) 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 $\bar{x}$

The estimated standard deviation of the sampling distribution of $\bar{x}$ when c3 is unknown: $SE = \frac{s}{\sqrt{n}}$.

One-sample t confidence interval formula

$\bar{x} \text{ ± } t^*\bigg(\frac{s}{\sqrt{n}}\bigg)$, 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, $ME = t^* \times \frac{s}{\sqrt{n}}$.

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 bc_1 - bc_2: ($\bar{x}_1 - \bar{x}_2$) \boldsymbol{\pm} t^* \times \sqrt\bigg(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\bigg), used when comparing two independent groups with unknown c3_1 and c3_2.

Standard error for difference of two means

SE = \sqrt\bigg(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\bigg), combining variability from both samples when estimating bc_1 - bc_2.

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 bc_1 - bc_2.