← Interval Estimates for Proportions › Margin of Error and Level of Confidence
Margin of Error and Level of Confidence
Key Terms
- Margin of error (ME)
- = z* × √(p̂(1−p̂)/n) — the half-width of the confidence interval
- ME decreases when n increases (more data → more precise)
- ME increases when confidence level increases (same data, more certainty → wider interval)
- To find the required sample size for a given ME: n ≥ (z*/ME)² × p̂(1−p̂)
- If p̂ is unknown, use p̂ = 0.5 (gives the largest/most conservative n)
- Always round n up to the nearest whole number
Margin of Error:
ME = z* × √(p̂(1−p̂)/n)
Required Sample Size:
n ≥ (z*)² × p̂(1−p̂) / ME²
Conservative (p̂ unknown): n ≥ (z*)² / (4 × ME²)
(Uses p̂(1−p̂) = 0.25 when p̂ = 0.5)
ME = z* × √(p̂(1−p̂)/n)
Required Sample Size:
n ≥ (z*)² × p̂(1−p̂) / ME²
Conservative (p̂ unknown): n ≥ (z*)² / (4 × ME²)
(Uses p̂(1−p̂) = 0.25 when p̂ = 0.5)
Worked Example: A researcher wants to estimate a population proportion with a margin of error of at most 0.04 at the 95% confidence level. How large a sample is needed if (a) p̂ is unknown, (b) a pilot study suggests p̂ ≈ 0.3?
(a) Unknown p̂: use p̂ = 0.5 (conservative)
n ≥ (1.96)² × 0.5 × 0.5 / (0.04)²
n ≥ 3.8416 × 0.25 / 0.0016 = 0.9604 / 0.0016 = 600.25
→ n = 601
(b) p̂ = 0.3:
n ≥ (1.96)² × 0.3 × 0.7 / (0.04)²
n ≥ 3.8416 × 0.21 / 0.0016 = 0.8067 / 0.0016 = 504.2
→ n = 505
(a) Unknown p̂: use p̂ = 0.5 (conservative)
n ≥ (1.96)² × 0.5 × 0.5 / (0.04)²
n ≥ 3.8416 × 0.25 / 0.0016 = 0.9604 / 0.0016 = 600.25
→ n = 601
(b) p̂ = 0.3:
n ≥ (1.96)² × 0.3 × 0.7 / (0.04)²
n ≥ 3.8416 × 0.21 / 0.0016 = 0.8067 / 0.0016 = 504.2
→ n = 505
Hot Tip: To halve the margin of error, you must quadruple the sample size (since ME ∝ 1/√n). This is why large, precise polls are expensive.
What is the Margin of Error?
The margin of error (ME) is the maximum expected difference between the sample proportion p̂ and the true population proportion p at a given confidence level. It is the half-width of the confidence interval:
ME = z* × √(p̂(1−p̂)/n)
A CI can be written as: p̂ − ME < p < p̂ + ME.
Factors Affecting the Margin of Error
Three factors determine ME:
- Sample size n: ME decreases as n increases. Doubling n reduces ME by a factor of √2 ≈ 1.41.
- Confidence level (z*): Higher confidence requires larger z*, which increases ME. You pay for certainty with precision.
- Sample proportion p̂: ME is largest when p̂ = 0.5. Extreme proportions (near 0 or 1) give smaller ME.
Choosing Sample Size
Often researchers need to decide on n before collecting data, to guarantee a target precision:
- Set ME = desired maximum error (e.g., 0.03)
- Choose confidence level (z* = 1.96 for 95%)
- Estimate p̂ if possible; otherwise use p̂ = 0.5
- Solve: n ≥ (z*)² × p̂(1−p̂) / ME²
- Round up to the nearest integer
Why Use p̂ = 0.5 When Unknown?
The product p(1−p) is maximised when p = 0.5, giving p(1−p) = 0.25. Using p̂ = 0.5 therefore produces the largest (most conservative) required sample size, guaranteeing the desired precision regardless of the actual p.
Strategy: In exam questions, always state which p̂ you are using in the sample size formula and explain your choice. If a pilot estimate is given, use it — it gives a smaller (but still adequate) sample size.
Mastery Practice
- Fluency A 95% CI for p is based on n = 400 and p̂ = 0.6. Calculate the margin of error.
- Fluency A researcher wants a margin of error of at most 0.05 with 95% confidence. No prior estimate of p is available. Find the minimum required sample size.
- Fluency Calculate the margin of error for a 99% CI with p̂ = 0.35 and n = 500.
- Fluency A 95% CI is reported as (0.44, 0.56). What is the margin of error and the sample proportion?
- Understanding A pollster currently uses n = 400 for 95% CIs. How large a sample would they need to halve the current margin of error?
- Understanding A pilot study finds p̂ ≈ 0.25. A researcher wants ME ≤ 0.03 at the 95% confidence level. Find the required n and compare it to the conservative estimate using p̂ = 0.5.
- Understanding Two surveys have identical n = 200 and p̂ = 0.5. Survey A reports a 90% CI and Survey B reports a 99% CI. Which has a larger margin of error? Calculate both and find the ratio of the margins.
- Understanding A TV network claims its news program is watched by 30% of households ± 3% (margin of error). If the 95% CI formula was used, back-calculate the approximate sample size used.
- Problem Solving A health researcher wants to estimate the proportion of adults with high blood pressure. From previous studies, p is thought to be near 0.22. The researcher wants a 99% CI with ME ≤ 0.025. Find the required sample size using (a) the prior estimate and (b) the conservative approach. How many extra participants does the conservative approach require?
- Problem Solving An election analyst wants to be 95% confident that their estimate of a candidate’s true proportion of support is within 2 percentage points of the actual value.
- (a) Find the minimum sample size needed (conservative).
- (b) If the analyst can only afford n = 600, what is the actual margin of error?
- (c) The analyst suggests increasing the confidence level to 99% while keeping n = 600. What is the new margin of error?