Chi-Square & Inference: every key term you need (+ practice quiz)

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Probability
• Random Variables & Distributions
• Sampling Distributions

Two-Sample t-Confidence Interval

(x̄₁ - x̄₂) ± t*·SE where SE = √(s₁²/n₁ + s₂²/n₂). Compares two population means. df approximated; often use smaller n-1.

Two-Sample t-test

H₀: μ₁ = μ₂ vs Hₐ: μ₁ ≠ μ₂. Test statistic: t = (x̄₁ - x̄₂)/√(s₁²/n₁ + s₂²/n₂). Compare to t-distribution.

Paired t-test

Compare dependent samples (before/after, matched pairs). Test: H₀: μ_d = 0. Use difference d = x₁ - x₂, treat as one-sample test.

Linear Regression Inference

Test H₀: slope β = 0 (no linear relationship). t = b/SE(b). If p < α, slope significantly ≠ 0.

Regression Slope CI

b ± t*·SE(b). Estimates true slope β. Interpretation: 95% CI for true change in y per unit x.

Regression Standard Error

SE = √(Σ(residuals)²/(n-2)). Measures typical distance points deviate from line. Smaller SE = better fit.

ANOVA (Analysis of Variance)

Tests if three or more means equal. H₀: μ₁ = μ₂ = μ₃. Test statistic F = MS_between / MS_within.

ANOVA Assumptions

Random samples from each group, independent observations, normal distributions within groups, equal variances. ANOVA robust to moderate violations.

Multiple Comparisons

If ANOVA rejects H₀, follow with pairwise comparisons (Tukey). Corrects for multiple testing (increases chance of false positive).

Effect Size (Cohen's d)

d = (x̄₁ - x̄₂)/s_pooled. Standardized difference. d<0.2 small, 0.2-0.8 medium, >0.8 large. Complements p-value.

Correlation & Causation

Strong correlation in observational data ≠ causation. Could be: reverse causation, confounding, lurking variable, coincidence.

Unit 8 Summary

Two-sample tests compare means. ANOVA extends to 3+ groups. Regression tests slope significance. Always consider practical significance beyond p-values.

Chi-Square Statistic

χ^2 = Σ (Observed - Expected)^2 / Expected, summed over all cells. Larger values mean observed counts are farther from what H0 predicts.

Goodness-of-Fit Test

One categorical variable, one sample. H0: the population distribution matches a claimed set of proportions. df = number of categories - 1.

GOF Expected Counts

Expected_i = n · p_i for each category. All expected counts must be at least 5 (some allow 80% ≥ 5, none < 1).

Test for Homogeneity

Two or more independent samples/populations (or treatment groups), one categorical variable. H0: the distribution of the variable is the same in each population.

Test for Independence

ONE sample, two categorical variables measured on each individual. H0: no association between the two variables in the population.

Homogeneity vs Independence Cue

How were data collected? Separate random samples from several groups → homogeneity. One random sample classified two ways → independence. Same mechanics, different hypotheses.

Two-Way Table Expected Count

Expected = (row total × column total) / grand total. Reflects what the cell would hold if H0 (no association / same distribution) were true.

Two-Way Table df

df = (rows - 1)(columns - 1). A 3×4 table has df = 6.

Chi-Square Distribution

Right-skewed, values ≥ 0, mean = df; becomes more symmetric as df grows. All chi-square tests are one-tailed to the right.

Chi-Square P-value

P = χ^2cdf(statistic, ∞, df). Large χ^2 → small P → evidence against H0.

Chi-Square Conditions

Random (samples or assignment), 10% if sampling without replacement, Large Counts: all EXPECTED counts ≥ 5.

Follow-Up Analysis

After rejecting H0, identify which cells contribute most to χ^2 and compare observed vs expected there to describe the nature of the association.

Chi-Square vs Two-Proportion z

For a 2×2 homogeneity table, χ^2 = z^2 and P-values match for a two-sided z test. Use z when a one-sided alternative is needed.

Chi-Square Uses Counts

Always compute χ^2 from raw counts, never from percentages; percentages destroy the sample size information.

Regression Slope Inference

Population model y = α + βx + ε. Test H0: β = 0 (no linear relationship) with t = b/SE_b, df = n - 2.

Conditions for Slope Inference (LINER)

Linear relationship, Independent observations (10%), Normal residuals at each x, Equal SD of residuals across x, Random sample/assignment.

Confidence Interval for Slope

b ± t* · SE_b with df = n - 2. Interpret: 'We are 95% confident that for each 1-unit increase in x, the mean y changes by between a and b units.'

Reading Slope Output

In computer output, the predictor row gives Coef = b, SE Coef = SE_b, T = b/SE_b, P = two-sided P-value for H0: β = 0. Halve P for a one-sided Ha.

Standard Error of Slope

SE_b = s / (s_x · sqrt(n-1)). Smaller with more spread in x, more data, or smaller residual SD s.

Residual Checks for Regression Inference

Residual plot: no pattern (linear), roughly constant spread (equal SD). Histogram or normal probability plot of residuals: roughly Normal.

Choosing the Right Test

Counts in categories → chi-square. Means → t. Proportions → z. Slope → t with df = n - 2. Identify the variable types first.

Chi-Square Cell Contribution

(O - E)^2/E for one cell. The largest contributions point to where the association lives, e.g. 'far more observed than expected.'