Confidence Interval for a Proportion
Estimate a population proportion — or the difference between two proportions — from sample counts. Enter the number of successes x and the sample size n; the result matches the TI-84 1-PropZInt and 2-PropZInt screens.
Planning a survey instead? The sample size calculator finds the n needed for a target margin of error. Part of the TI-84 statistics functions guide.
Related Calculators
Z-Test Calculator
Run one-sample, two-sample, and proportion z-tests with z statistics, p-values, and critical values.
Margin of Error Calculator
Measure sampling error for proportions and means at common confidence levels.
Sample Size Calculator
Plan surveys with a confidence level, margin of error, and finite population correction.
Formulas
One proportion: p̂ ± z* · √(p̂(1 − p̂) / n)
Two proportions: (p̂₁ − p̂₂) ± z* · √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂)
z* = invNorm((1 + C) / 2): 90% → 1.645, 95% → 1.960, 99% → 2.576
How to Use 1-PropZInt on a TI-84
- Press STAT → TESTS.
- Choose A:1-PropZInt (or B:2-PropZInt for two proportions).
- Enter x (a whole-number count, not a percentage), n, and C-Level.
- Highlight Calculate and press ENTER. The screen shows the interval, p̂, and n.
Worked Example: A Poll
In a poll of 1,000 voters, 412 support a proposal. Find a 95% interval.
- p̂ = 412 / 1000 = 0.412.
- SE = √(0.412 × 0.588 / 1000) = 0.01556; margin of error = 1.96 × 0.01556 = 0.0305.
- Interval: (0.3815, 0.4425) — between about 38% and 44% support.
That ±3 percentage points is the "margin of error" quoted in news reports. See the margin of error calculator for how it depends on n.
Worked Example: Two Proportions
120 of 400 customers who saw design A clicked, versus 90 of 380 who saw design B. Find a 95% interval for p_A − p_B.
- p̂_A = 0.3, p̂_B = 0.2368, difference = 0.0632.
- Interval: (0.0012, 0.1252).
The whole interval is above 0, so design A's click rate is significantly higher at the 5% level — though the difference could be as small as 0.1 percentage points.
Frequently Asked Questions
What conditions are needed for a proportion confidence interval?
A random sample, independent observations (the sample is less than 10% of the population when sampling without replacement), and at least 10 successes and 10 failures in each sample so the normal approximation holds.
Why does the TI-84 give an error when I enter a percentage?
1-PropZInt needs x as a whole-number count. If a report says 41.2% of 1,000 people, enter x = 412, not 0.412 or 41.2. This calculator enforces the same rule.
How do I make the interval narrower?
Increase the sample size or lower the confidence level. The margin of error shrinks with 1/√n, so halving it requires about four times as many observations.
Is this the same as the Wald interval?
Yes. 1-PropZInt uses the standard (Wald) interval p̂ ± z*·√(p̂(1 − p̂)/n), which is what introductory courses and the TI-84 use. For very small samples or proportions near 0 or 1, methods such as the Wilson interval are more accurate.
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