Triola Elementary Statistics · 11th Edition
Chapter 7: Estimating Parameters & Sample Sizes
Every key formula from Triola Elementary Statistics Chapter 7, in one searchable page. Click a card to study it — worked examples included.
7.2 · CI FOR A PROPORTION
Point estimate
p̂ = x/n is the best point estimate of p (unbiased)
Notation
q̂ = 1 − p̂; zα/2 = z with area α/2 in the right tail.
Margin of error (Formula 7-1)
E = zα/2 · √(p̂q̂n)
Confidence interval
p̂ − E < p < p̂ + E
Requirements: simple random sample; binomial conditions; np̂ ≥ 5 and nq̂ ≥ 5 (≥ 5 successes and ≥ 5 failures).
Common z* values
90% → 1.645 · 95% → 1.96 · 99% → 2.575
WORKED EXAMPLE
Pew poll: n = 1501, p̂ = 0.70, 95%:
E = 1.96·√(0.7·0.3/1501) = 0.023 ⇒
0.677 < p < 0.723.
E = 1.96·√(0.7·0.3/1501) = 0.023 ⇒
0.677 < p < 0.723.
KEY NOTES
- Say: “We are 95% confident the interval contains p” — not “the probability p is inside is 95%”.
7.2 · SAMPLE SIZE (p)
n = [zα/2]² p̂q̂E² (estimate known)
n = [zα/2]² · 0.25E² (no estimate — conservative)
KEY NOTES
- Always round n UP to the next whole number.
- n does not depend on the population size N.
WORKED EXAMPLE
95% confidence, E = 0.03, no prior estimate:
n = 1.96² · 0.25 / 0.03² = 1067.11 ⇒ n = 1068.
n = 1.96² · 0.25 / 0.03² = 1067.11 ⇒ n = 1068.
7.3 · CI FOR A MEAN — σ KNOWN
x̄ − E < μ < x̄ + E, E = zα/2 · σ√n
Requirements: simple random sample; normal population or n > 30.
WORKED EXAMPLE
n = 36, x̄ = 100, σ = 15, 95%:
E = 1.96 · 15/6 = 4.9 ⇒ 95.1 < μ < 104.9.
E = 1.96 · 15/6 = 4.9 ⇒ 95.1 < μ < 104.9.
KEY NOTES
- x̄ is the best point estimate of μ: it is the midpoint of any CI, and E is the half-width.
7.3 · MEAN CI: σ UNKNOWN
E = tα/2 · s√n (df = n − 1)
Use Table A-3, column “Area in Two Tails”. t is wider than z because s estimates σ.
WORKED EXAMPLE
n = 16, x̄ = 80, s = 10, 95% (df = 15):
t0.025 = 2.131; E = 2.131 · 10/4 = 5.33 ⇒
74.67 < μ < 85.33.
t0.025 = 2.131; E = 2.131 · 10/4 = 5.33 ⇒
74.67 < μ < 85.33.
KEY NOTES
- σ known → z (even for small n). σ unknown → t. Never plug s into the z formula.
7.4 · SAMPLE SIZE FOR A MEAN
n = (zα/2 · σE)2 (Formula 7-4)
σ unknown? Estimate σ ≈ range/4. Round n up.
WORKED EXAMPLE
95% confidence, σ = 12, E = 2:
n = (1.96 · 12/2)² = 138.30 ⇒ n = 139.
n = (1.96 · 12/2)² = 138.30 ⇒ n = 139.
7.5 · CI FOR VARIANCE & STD DEV
χ² = (n − 1)s²σ² (df = n − 1)
Chi-square: not symmetric, never negative.
Confidence interval
(n − 1)s²χ²R < σ² < (n − 1)s²χ²L
For σ: take √ of both limits. Population must be normal (strict!).
WORKED EXAMPLE
n = 30, s = 0.15, 95% (df = 29):
χ²L = 16.047, χ²R = 45.722 ⇒
0.0143 < σ² < 0.0407; 0.119 < σ < 0.202.
χ²L = 16.047, χ²R = 45.722 ⇒
0.0143 < σ² < 0.0407; 0.119 < σ < 0.202.
WATCH OUT!
Mistakes that cost points
- CI uses zα/2: for 95% look up 0.9750, not 0.95
- Sample size: ALWAYS round up — rounding down misses the target E
- σ known → z; σ unknown → t (df = n−1)
- χ² CI: the LEFT critical χ²L goes in the RIGHT denominator
- χ² methods need a NORMAL population even when n is large
- “95% confident the interval contains μ” ≠ “95% chance μ is inside”
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