Triola Elementary Statistics · 11th Edition

Chapter 13: Nonparametric Tests

Every key formula from Triola Elementary Statistics Chapter 13, in one searchable page. Click a card to study it — worked examples included.

13.2  ·  SIGN TEST
Tests a claim about the median (or matched pairs, or two-category data). Drop ties.
x = # of the less frequent sign,  n = total # of signs
Test statistic
n ≤ 25: test stat = x (Table A-7)
n > 25: z = (x + 0.5) − n/2√n/2 (Table A-2)
WORKED EXAMPLE
Freshman weights: 7 −, 2 +, 1 tie ⇒ n = 9, x = 2;
critical value = 1 (Table A-7) ⇒ fail to reject.
XSORT: n = 726, x = 58 ⇒ z = −22.60, critical −1.645 ⇒ reject.
13.3  ·  WILCOXON SIGNED-RANK
Matched pairs. Rank |d| (ties: mean rank); discard d = 0 pairs.
T = smaller of (∑ positive ranks, |∑ negative ranks|)
Test statistic
n ≤ 30: test stat = T (Table A-8)
n > 30: z = T − n(n+1)/4√[n(n+1)(2n+1)/24]
WORKED EXAMPLE
Freshman weights: rank sums 5 and 40 ⇒ T = 5, n = 9;
critical = 6 (Table A-8, two-tailed) ⇒ reject (small T rejects).
13.4  ·  WILCOXON RANK-SUM
Two INDEPENDENT samples. Combine, rank all values, then R = rank sum of Sample 1.
z = R − μRσR,  μR = n1(n1+n2+1)2
σR = √n1n2(n1+n2+1)12
Requires each sample > 10 values.
WORKED EXAMPLE
Braking distances, 4-cyl vs 6-cyl: R = 180.5, n1 = 13, n2 = 12 ⇒
μR = 169, σR = 18.385, z = 0.63 ⇒ fail to reject.
WATCH OUT!
Mistakes that cost points
  • Sign test: EXCLUDE ties — n shrinks
  • Wilcoxon signed-rank: rank |d|, then re-attach signs; T is the SMALLER sum
  • Rank-sum: combine BOTH samples before ranking
  • Watch the sample-size cutoffs: 25 / 30 / 10 / 20 / 5
  • Nonparametric ≠ no parameters — many tests are about the MEDIAN
13.5  ·  KRUSKAL-WALLIS
H0: samples come from populations with equal medians (3+ independent samples)
H = 12N(N+1)[R1²/n1 + R2²/n2 + … + Rk²/nk] − 3(N+1)
Ri = rank sum of sample i; right-tailed, df = k − 1 (Table A-4).
KEY NOTES
  • Each sample must have ≥ 5 observations.
  • The rank version of the ANOVA F test.
WORKED EXAMPLE
Car-crash deceleration, 3 sizes: H = 5.774;
critical = 5.991 (df = 2, α = 0.05) ⇒ fail to reject equal medians.
13.6  ·  RANK CORRELATION
Spearman rs: correlation using ranks instead of values. H0: rs = 0.
No ties
rs = 1 − 6∑d²n(n² − 1),  d = rank difference
Ties
Use Formula 10-1 with the ranks in place of x, y.
Critical values
n ≤ 30: Table A-9. n > 30: ±z/√(n−1).
WORKED EXAMPLE
University rankings: n = 8, ∑d² = 156 ⇒
rs = 1 − 936/504 = −0.857; critical ±0.738 ⇒ reject.
13.7  ·  RUNS TEST
Tests randomness of a sequence. G = number of runs; n1, n2 = counts of the two characteristics.
Test statistic
Small samples (n1, n2 ≤ 20): test stat = G (Table A-10)
Large: z = G − μGσG,  μG = 2n1n2n1+n2 + 1
KEY NOTES
  • Reject randomness if runs are very few OR very many.
  • Order matters — frequencies do NOT (3 men, 20 women can still be “random” order).
WORKED EXAMPLE
Gender sequence: n1 = 7, n2 = 8, G = 6;
critical 4 and 13 ⇒ fail to reject randomness.
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