Triola Elementary Statistics · 11th Edition
Chapter 11: Goodness-of-Fit & Contingency Tables
Every key formula from Triola Elementary Statistics Chapter 11, in one searchable page. Click a card to study it — worked examples included.
11.2 · GOODNESS-OF-FIT
Tests whether observed counts fit a claimed distribution.
Test statistic
χ² = ∑(O − E)²E
O = observed count, E = expected count, k = categories.
Critical values
df = k − 1, always right-tailed (Table A-4)
KEY NOTES
- Requirements: random sample, frequency counts, E ≥ 5 for EVERY category.
- Expected from the claim: E = np for each category.
11.2 · WORKED EXAMPLE
WORKED EXAMPLE
Last digits of 80 weights — are they equally likely?
H0: p0 = p1 = … = p9; E = 8 each.
χ² = 11.250; critical = 16.919 (df = 9, α = 0.05).
Fail to reject — digits look uniform.
H0: p0 = p1 = … = p9; E = 8 each.
χ² = 11.250; critical = 16.919 (df = 9, α = 0.05).
Fail to reject — digits look uniform.
χ² is never negative; larger values = bigger disagreement between O and E.
11.3 · CONTINGENCY TABLES
Tests independence of the row and column variables.
H0: row and column variables are independent.
Expected frequency
E = (row total)(column total)grand total
Test
χ² = ∑(O − E)²E, df = (r − 1)(c − 1)
Right-tailed; E ≥ 5 in EVERY cell.
11.3 · WORKED EXAMPLE
WORKED EXAMPLE
Echinacea experiment (infection × treatment), 207 subjects:
First cell O = 88: E = (178)(103)/207 = 88.570.
χ² = 7.885; critical = 7.815 (df = 1, α = 0.05).
Reject H0 — infection depends on treatment.
First cell O = 88: E = (178)(103)/207 = 88.570.
χ² = 7.885; critical = 7.815 (df = 1, α = 0.05).
Reject H0 — infection depends on treatment.
Under independence, P(A ∩ B) = P(A)·P(B) — that is exactly what the E formula builds.
11.3 · TEST OF HOMOGENEITY
Claim: different populations have the same proportions of some characteristic.
KEY NOTES
- Use the SAME notation, requirements, test statistic, critical value, and procedure as the test of independence.
- Difference is the DESIGN: separate samples from different populations (e.g. men vs women interviewed).
H0: the populations are homogeneous
WATCH OUT!
Mistakes that cost points
- E comes from the CLAIM (gof: np; table: row·col/total) — never from O
- Every expected frequency must be ≥ 5 (observed O can be anything)
- Counts only — not means, percentages, or measurements
- χ² tests are ALWAYS right-tailed
- Critical values from Table A-4, not the t table
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