Inference for Means: every key term you need (+ practice quiz)
37 flashcard terms for AP Statistics Unit 6, written to match the course framework. Read them here, drill them as flashcards, or take the 28-question quiz. Free, no account needed.
Range of plausible values for population parameter. 95% CI means: if we repeated sampling infinitely, 95% of intervals contain true μ.
Confidence Level
Probability that confidence interval method produces interval containing parameter. 90%, 95%, 99% common. Higher % = wider interval.
Margin of Error
Half-width of confidence interval; distance from estimate to endpoint. ME = t*·SE. Larger n or lower confidence = smaller ME.
CI for Population Mean (σ known)
x̄ ± z*·(σ/√n) where z* depends on confidence level (1.96 for 95%). Rarely used (σ usually unknown).
CI for Population Mean (σ unknown)
x̄ ± t*·(s/√n) where t* from t-distribution with df=n-1. s = sample standard deviation. Most practical scenario.
Hypothesis Test Setup
H₀ (null): status quo, no effect, parameter = specified value. Hₐ (alternative): claim to test. Test whether data contradicts H₀.
One-tailed vs Two-tailed Tests
One-tailed: Hₐ says > or <. Two-tailed: Hₐ says ≠. Two-tailed more conservative; splits α between tails.
P-value Definition
Probability of observing test statistic as extreme or more extreme assuming H₀ true. Small p-value = strong evidence against H₀.
Significance Level α
Threshold for rejecting H₀; typically 0.05. If p-value < α, reject H₀ (statistically significant result).
Type I Error
Reject H₀ when true (false positive). Probability = α. Example: conclude medicine works when it doesn't.
Type II Error
Fail to reject H₀ when false (false negative). Probability = β. Example: conclude medicine doesn't work when it does.
Power
Power = 1 - β; probability of rejecting false H₀. Higher power = better test. Increases with larger n, larger effect, larger α.
t-test for Population Mean
Test H₀: μ = μ₀. Test statistic: t = (x̄ - μ₀)/(s/√n). Compare to t-distribution with df = n-1.
Assumptions for t-test
Data from random sample, observations independent, population approximately normal (or large n). Robust to normality if n large.
Unit 6 Summary
Confidence intervals estimate parameters with specified confidence. Hypothesis tests assess evidence against null hypothesis using p-values and significance levels.
One-Sample t Interval
x̄ ± t* · s/sqrt(n) with df = n - 1. Conditions: random, 10% (if sampling without replacement), Normal/large sample.
Normal/Large Sample Condition
Population Normal, or n ≥ 30, or for smaller n a graph of the sample data shows no strong skew or outliers.
Robustness of t Procedures
t procedures tolerate moderate non-Normality, especially for large n, but are sensitive to outliers and strong skew in small samples.
One-sample and paired: n - 1. Two-sample: technology's fractional df (Welch), or conservative min(n1-1, n2-1).
Interpreting a Confidence Interval
'We are 95% confident the interval from a to b captures the true mean [context].' Never say the probability μ is in this interval is 95%.
Interpreting Confidence Level
'If we took many random samples and built an interval from each, about 95% of those intervals would capture the true mean.' It describes the method, not one interval.
Margin of Error Drivers
Increases with confidence level and with s; decreases with sample size. Does not account for bias, nonresponse, or bad design.
Sample Size for a Mean
n ≥ (z*·σ/ME)^2, using z* since t* depends on unknown n; round up. Uses a guessed σ from a pilot study or prior data.
One-Sample t Test
t = (x̄ - μ0)/(s/sqrt(n)), df = n - 1. P-value from tcdf. Same conditions as t interval.
Paired t Procedures
When observations come in natural pairs (before/after, twins, two measurements per subject), analyze the differences with one-sample t methods.
Paired vs Two-Sample Decision
Ask: is each observation in group 1 linked to a specific observation in group 2? Yes → paired. Independent groups → two-sample.
Two-Sample t Interval
(x̄1 - x̄2) ± t* · sqrt(s1^2/n1 + s2^2/n2). Conditions apply to both samples; do NOT pool unless told variances equal (AP does not pool).
Rejecting a true H0. Probability = α. Consequence: acting on an effect that isn't real.
Type II Error
Failing to reject a false H0. Probability = β. Consequence: missing a real effect.
Power
1 - β: probability of correctly rejecting a false H0. Increases with n, α, effect size, and decreases with σ.
Significance vs Confidence Interval Duality
A two-sided test at level α rejects H0: μ = μ0 exactly when a (1-α) confidence interval excludes μ0.
Four-Step Inference
State (parameter, hypotheses, α), Plan (procedure, conditions), Do (statistic, P-value or interval), Conclude (in context, linked to H0/Ha).
Conclusion Wording
'Because P = 0.03 < α = 0.05, we reject H0. We have convincing evidence that [Ha in context].' Or 'fail to reject... do not have convincing evidence.' Never 'accept H0.'
P-value Interpretation
'Assuming H0 is true, the probability of getting a sample statistic at least as extreme as ours is [P].' Not the probability H0 is true.
Effect of Outliers on t
An outlier inflates s and drags x̄, weakening evidence and widening intervals; check with a boxplot before proceeding on small samples.