Triola Elementary Statistics · 11th Edition
Chapter 9: Inferences from Two Samples
Every key formula from Triola Elementary Statistics Chapter 9, in one searchable page. Click a card to study it — worked examples included.
9.2 · TWO PROPORTIONS
Notation
p̂1 = x1/n1, p̂2 = x2/n2; q̂ = 1 − p̂.
Requirements: SRS; independent samples; np ≥ 5, nq ≥ 5 for each sample.
Requirements: SRS; independent samples; np ≥ 5, nq ≥ 5 for each sample.
Confidence interval (do NOT pool)
(p̂1−p̂2) − E < p1−p2 < (p̂1−p̂2) + E
E = zα/2 · √(p̂1q̂1n1 + p̂2q̂2n2)
Hypothesis test, H0: p1 = p2 (pool!)
z = (p̂1−p̂2) − 0√(p̄q̄/n1 + p̄q̄/n2)
p̄ = x1 + x2n1 + n2, q̄ = 1 − p̄
WORKED EXAMPLE
n1 = n2 = 100, x1 = 60, x2 = 45; H1: p1 ≠ p2, α = 0.05:
p̄ = 105/200 = 0.525; SE = √(0.525·0.475·0.02) = 0.0706;
z = 0.15/0.0706 = 2.12; P-value = 0.0337 < 0.05 ⇒ reject H0.
p̄ = 105/200 = 0.525; SE = √(0.525·0.475·0.02) = 0.0706;
z = 0.15/0.0706 = 2.12; P-value = 0.0337 < 0.05 ⇒ reject H0.
Confidence interval
(x̄1−x̄2) − E < μ1−μ2 < (x̄1−x̄2) + E
E = tα/2 · √(s1²n1 + s2²n2)
Hypothesis test
t = (x̄1−x̄2) − (μ1−μ2)√(s1²/n1 + s2²/n2)
(μ1−μ2 = 0 under H0.) df = smaller of n1−1, n2−1 (conservative).
Requirements: σ1, σ2 unknown, not assumed equal; independent SRS; n1, n2 > 30 or both populations normal.
Requirements: σ1, σ2 unknown, not assumed equal; independent SRS; n1, n2 > 30 or both populations normal.
WORKED EXAMPLE
x̄1 = 100, s1 = 12, n1 = 25; x̄2 = 94, s2 = 10, n2 = 20; two-tailed, α = 0.05:
SE = √(144/25 + 100/20) = 3.28; t = 6/3.28 = 1.83;
df = 19; critical ±2.093 ⇒ fail to reject H0.
SE = √(144/25 + 100/20) = 3.28; t = 6/3.28 = 1.83;
df = 19; critical ±2.093 ⇒ fail to reject H0.
INDEPENDENT vs DEPENDENT
Which method?
- Same subjects measured twice (before/after)? → matched pairs (§9.4)
- Natural pairing (twins, matched by age)? → matched pairs
- Two separate groups, no pairing? → independent samples (§9.3)
- Comparing proportions of two groups? → two proportions (§9.2)
- Comparing spread of two groups? → F test (§9.5)
Notation
d = individual difference (one per pair); d̄ = mean of the d‘s; sd = SD of the d‘s; n = # of pairs.
Confidence interval
d̄ − E < μd < d̄ + E, E = tα/2 · sd√n
Hypothesis test
t = d̄ − μdsd/√n (df = n − 1)
Requirements: dependent data; SRS; n > 30 or differences ~normal (robust — check outliers).
WORKED EXAMPLE
n = 8 pairs, d̄ = 2.5, sd = 4.0; claim μd = 0, two-tailed, α = 0.05:
t = 2.5/(4/√8) = 1.77; df = 7; critical ±2.365 ⇒ fail to reject H0.
t = 2.5/(4/√8) = 1.77; df = 7; critical ±2.365 ⇒ fail to reject H0.
KEY NOTES
- Paired data = ONE sample of differences — never run it as two independent samples.
9.5 · F TEST FOR VARIANCES
F = s1²s2² (put the larger variance on top)
df1 = n1 − 1 (numerator), df2 = n2 − 1 (denominator); Table A-5.
F distribution: not symmetric, never negative, shape depends on both df.
F distribution: not symmetric, never negative, shape depends on both df.
Requirements
Simple random samples; both populations normal — the F test is not robust.
WORKED EXAMPLE
s1 = 4.2, n1 = 16; s2 = 3.1, n2 = 21; H1: σ1 > σ2, α = 0.05:
F = 17.64/9.61 = 1.836; df = (15, 20);
critical F0.05 = 2.2033; 1.836 < 2.2033 ⇒ fail to reject H0.
F = 17.64/9.61 = 1.836; df = (15, 20);
critical F0.05 = 2.2033; 1.836 < 2.2033 ⇒ fail to reject H0.
KEY NOTES
- Larger s² on top means only the right-tail critical value is needed, even for two-tailed tests.
WATCH OUT!
Mistakes that cost points
- Paired design → work with the differences d (one sample, n pairs)
- Two-proportion test pools p̄; the CI never pools
- Two-means df = smaller of n1−1, n2−1 (conservative)
- F test: larger s² on top; both populations must be normal
- Tests assume H0: p1 = p2, μ1 = μ2, σ1² = σ2² (difference = 0)
- Independence first: check the design before choosing §9.3 vs §9.4