What is the formula for the proportion Z-test?

Z = (p̂ − p₀) / √[p₀(1−p₀)/n], where p̂ is the sample proportion, p₀ is the null-hypothesized proportion, and n is the sample size.

When can I use a Z-test for proportions?

When the normal approximation conditions are met: np₀ ≥ 10 AND n(1−p₀) ≥ 10. Always check these conditions using p₀, not p̂ — using p̂ is the Ghost Proportion Error. If either condition fails, use an exact binomial test instead.

What does the p-value mean in a proportion test?

The p-value is the probability of observing a test statistic as extreme as the one computed, assuming the null hypothesis is true. A p-value of 0.03 means there is only a 3% chance of seeing data this extreme if H₀ were true. It is NOT the probability that H₀ is true.

What is proportion hypothesis testing NOT used for?

Proportion hypothesis testing is NOT for continuous data (use a t-test for means), NOT for paired observations (use McNemar's Test), NOT for 3+ groups (use Chi-Square), and a significant result does NOT prove the alternative hypothesis — it only provides evidence against H₀.

What is the difference between one-tailed and two-tailed proportion tests?

A two-tailed test asks whether the proportion differs in either direction (H₁: p ≠ p₀). A one-tailed test specifies direction (H₁: p > p₀ or H₁: p < p₀). The choice must be made before data collection — switching after seeing results is a form of p-hacking called HARKing.

Proportion Hypothesis Testing: Complete Guide with Formulas [2026]

Q: What is proportion hypothesis testing?

Proportion hypothesis testing is a statistical method that determines whether an observed sample proportion significantly differs from a hypothesized population proportion. Using the Z-distribution as a normal approximation to the binomial, it calculates a test statistic, compares it to a critical value, and produces a p-value to either reject or retain the null hypothesis.

What Is Proportion Hypothesis Testing?

Definition — Proportion Hypothesis Test

Proportion hypothesis testing is a statistical method that determines whether an observed sample proportion significantly differs from a hypothesized population proportion. Using the Z-distribution as a normal approximation to the binomial, it calculates a test statistic, compares it to a critical value, and produces a p-value to either reject or retain the null hypothesis.

Z = (p̂ − p₀) / √[p₀(1−p₀)/n]

At its core, proportion hypothesis testing answers one question: is the gap between what you observed (p̂) and what you expected (p₀) large enough to be statistically convincing, or is it the kind of gap that appears routinely by chance? The Z-statistic measures this gap in standard error units. The p-value converts that measurement into a probability.

This framework sits within the broader field of hypothesis testing covered at Statistics Fundamentals, alongside the one-sample t-test for means. The distinction between the two is fundamental: proportions arise from binary outcomes (yes/no, pass/fail, click/no-click), while means arise from continuous measurements. The binomial distribution governs the data-generation mechanism here, and the Z-test works because of a normal approximation justified by the Central Limit Theorem.

⚡ Quick Reference — Proportion Test Key Facts

Formula: Z = (p̂ − p₀) / √[p₀(1−p₀)/n] — uses p₀ in the standard error, not p̂
Condition: np₀ ≥ 10 AND n(1−p₀) ≥ 10 — check against p₀, not p̂ (Ghost Proportion Check)
Decision rule: Reject H₀ when p-value ≤ α; never write "accept H₀"
Critical values: ±1.960 (two-tailed, α=0.05); ±1.645 (one-tailed, α=0.05)
If conditions fail: Use Exact Binomial Test, not the Z-test

±1.960

Critical Z at α=0.05 (two-tailed)

np₀≥10

Required condition to use Z-test

p₀

Always in SE denominator — never p̂

Steps in the PRO-7 Protocol

What Proportion Hypothesis Testing Is NOT

Knowing where a test applies is as important as knowing how to compute it. Proportion hypothesis testing has firm boundaries, and violating them produces confident-looking answers to the wrong questions.

Scenario	Correct Tool?	Use Instead
Binary outcome (yes/no), 1 group, vs. benchmark	✅ YES	One-proportion Z-test
Continuous outcome (weight, salary, temperature)	❌ NO	One-sample t-test or Z-test for means
Comparing proportions between two independent groups	✅ YES (different formula)	Two-proportion Z-test
Comparing means between two groups	❌ NO	Two-sample t-test
Paired/repeated measurements on same subjects	❌ NO	McNemar's Test
Three or more proportion categories	❌ NO	Chi-Square Test of Homogeneity
Small sample where np₀ < 10	❌ NO	Fisher's Exact Test or Exact Binomial
Proving the alternative hypothesis is true	⚠️ MISCONCEPTION	Significant result only disproves H₀

⚖️

The Verdict Analogy

A significant proportion test is a verdict against the null, not a proof of the alternative. A courtroom acquittal doesn't prove innocence — it proves insufficient evidence of guilt. Proportion hypothesis testing operates by the same rule: rejecting H₀ (p = 0.50) means your sample is inconsistent with a fair coin, not that you've proven the coin is biased at any specific rate.

The Proportion Z-Test Formula — With Semantic Variable Keys

Each formula below is paired with a Semantic Variable Key: a structured table defining every symbol in plain language. This format ensures you know not just what to compute, but what each component represents statistically.

One-Proportion Z-Test Statistic

One-Proportion Z-Test — Test Statistic

Z = (p̂ − p₀) / √[p₀(1 − p₀) / n]

Use when outcome is binary and normal approximation conditions are met

p̂ = observed sample proportion p₀ = null-hypothesized proportion n = sample size Z = standard errors from null value

Symbol	Name	Plain-Language Definition	Valid Range
Z	Test Statistic	How many standard errors the observed proportion sits from the null-hypothesized value	(−∞, +∞)
p̂	Sample Proportion (Observed)	The fraction of successes in the sample: x divided by n. This is what you measured.	[0, 1]
p₀	Null Hypothesis Proportion (Assumed)	The population proportion claimed by H₀. This is the value being challenged.	[0, 1]
n	Sample Size	Total number of independent observations collected in the sample	Positive integer
p₀(1−p₀)	Null Variance	Variance of a single Bernoulli trial when p = p₀	[0, 0.25]
√[p₀(1−p₀)/n]	Standard Error (under H₀)	Expected standard deviation of p̂ across repeated samples, assuming H₀ is true. Uses p₀, not p̂.	[0, 0.5]

⚠️

Why p₀ — Not p̂ — Appears in the Standard Error

This is the most misunderstood formula detail. We compute the test statistic under the assumption that H₀ is true. The question being answered is: "If p₀ were the real proportion, how variable would p̂ be?" Using p̂ in the denominator answers a different question — it estimates variability under the alternative hypothesis, producing a test statistic that does not follow the null distribution. That is the Ghost Proportion Error.

Standard Error — Two Versions

Property	SE under H₀ (for Z-test)	SE observed (for Confidence Interval)
Formula	`√[p₀(1−p₀)/n]`	`√[p̂(1−p̂)/n]`
Which proportion	p₀ — the null value	p̂ — the observed value
When to use	Computing the Z-test statistic	Building a confidence interval
Assumes	H₀ is true	The sample reflects the population
Ghost Proportion Error risk	High — students often substitute p̂	None — p̂ is correct here

Confidence Interval for a Proportion

95% Confidence Interval — Proportion

CI = p̂ ± z_α/2 · √[p̂(1 − p̂) / n]

p̂ = center of the interval (sample proportion) z_α/2 = 1.960 for 95% confidence √[p̂(1−p̂)/n] = margin of error denominator

A confidence interval and a two-tailed hypothesis test are mathematically dual. If the hypothesized value p₀ falls outside the 95% CI, the Z-test rejects H₀ at α = 0.05. The CI adds practical value by showing the magnitude of the effect — something a binary reject/fail-to-reject decision cannot convey. The full framework for interval estimation is covered in the confidence intervals guide.

Critical Z-Values Reference

Confidence Level	α	Test Type	Critical Value (z*)
90%	0.10	Two-tailed	±1.645
95%	0.05	Two-tailed	±1.960
99%	0.01	Two-tailed	±2.576
90%	0.10	One-tailed	±1.282
95%	0.05	One-tailed	±1.645
99%	0.01	One-tailed	±2.326

If/Then Decision Logic — Which Test to Use

Selecting the correct proportion test requires a sequential check through four conditions. Skipping any one step is the root cause of the most common errors in proportion testing.

IF outcome variable is continuous (height, weight, score)

→ DO NOT USE proportion test

→ USE one-sample t-test or Z-test for means

IF outcome variable is binary (yes/no, pass/fail, click/no-click)

→ Continue to next condition

IF comparing to a single known benchmark p₀

→ Candidate: One-Proportion Z-Test

CHECK: np₀ ≥ 10 AND n(1−p₀) ≥ 10 (use p₀, NOT p̂)

IF both pass → USE One-Proportion Z-Test (PRO-7 Protocol)

IF either fails → USE Exact Binomial Test

IF comparing two independent groups

→ Candidate: Two-Proportion Z-Test

CHECK: all four: n₁p̂_pool ≥ 10, n₁(1−p̂_pool) ≥ 10, n₂p̂_pool ≥ 10, n₂(1−p̂_pool) ≥ 10

IF all pass → USE Two-Proportion Z-Test

IF any fail → USE Fisher's Exact Test

IF three or more groups → USE Chi-Square Test of Homogeneity

IF paired/dependent observations → USE McNemar's Test

Scenario	Test	Key Condition
Binary, 1 group, large n, vs. benchmark	One-Proportion Z-Test	np₀ ≥ 10 AND n(1−p₀) ≥ 10
Binary, 1 group, small n	Exact Binomial Test	np₀ < 10
Binary, 2 independent groups, large n	Two-Proportion Z-Test	All four pooled-proportion conditions ≥ 10
Binary, 2 independent groups, small n	Fisher's Exact Test	Any pooled condition < 10
Binary, paired groups	McNemar's Test	Within-subject design
Binary, 3+ groups	Chi-Square Homogeneity	Expected counts ≥ 5 per cell
Continuous, 1 group	One-Sample t-Test	σ unknown; use sample SD

The PRO-7 Protocol — Step-by-Step Proportion Testing

The PRO-7 Protocol is Statistics Fundamentals' seven-step framework for a valid proportion hypothesis test. Each step corresponds to a distinct statistical decision. Following the sequence prevents the most frequent errors, including the Ghost Proportion Error at Step 2 and the "Accept H₀" wording mistake at Step 6.

Postulate

State the Hypotheses

Define H₀: p = p₀ and H₁ with explicit directionality — two-tailed (H₁: p ≠ p₀), upper one-tailed (H₁: p > p₀), or lower one-tailed (H₁: p < p₀). Lock this in writing before collecting data. Selecting direction after observing results is HARKing (Hypothesizing After Results are Known) and artificially inflates statistical significance.

Restrict

Ghost Proportion Check — Normal Approximation

Compute np₀ and n(1−p₀). Both must be ≥ 10. Use p₀ here, not p̂ — checking with p̂ produces a false pass that allows an invalid test to proceed with misplaced confidence. If either condition fails, stop and use an Exact Binomial Test.

np₀ ≥ 10 AND n(1−p₀) ≥ 10

Observe

Compute the Sample Proportion

Calculate p̂ = x / n, where x is the count of observed successes and n is the total sample size. This is the only step where p̂ is computed. It represents what you actually measured — not an assumption.

p̂ = x / n

Operationalize

Calculate the Standard Error

Use p₀ in the formula, not p̂. The standard error here estimates how much p̂ would vary across repeated samples if H₀ were true. Substituting p̂ — the Ghost Proportion Error — estimates variability under a different assumption and corrupts the test statistic.

SE = √[p₀(1−p₀) / n]

Quantify

Compute Z and Find the P-Value

Z = (p̂ − p₀) / SE. Then convert to a p-value using the standard normal table or software. For two-tailed: p-value = 2 × P(Z > |z|). For upper one-tailed: p-value = P(Z > z). For lower one-tailed: p-value = P(Z < z).

Z = (p̂ − p₀) / SE

Rule

Make the Statistical Decision

Compare the p-value to the pre-specified significance level α. If p-value ≤ α: Reject H₀. If p-value > α: Fail to reject H₀. The phrase "Accept H₀" is never correct in frequentist statistics — absence of evidence is not evidence of absence.

Report

State the Conclusion in Context

Translate the statistical decision back into the original problem domain. Use the APA-style template: "There [is / is not] sufficient statistical evidence at the α = [value] significance level to conclude that the population proportion of [event] [differs from / exceeds / falls below] [p₀] (Z = [value], p = [value])."

Worked Examples Using the PRO-7 Protocol

Example 1 — Coin Fairness Test (Two-Tailed)

PRO-7 Worked Example 1 — Two-Tailed Test

A coin is flipped 200 times and lands heads 118 times. At α = 0.05, is there sufficient evidence to conclude the coin is unfair?

Postulate: H₀: p = 0.50 (coin is fair). H₁: p ≠ 0.50 (two-tailed). α = 0.05.

Ghost Proportion Check: np₀ = 200 × 0.50 = 100 ✅. n(1−p₀) = 200 × 0.50 = 100 ✅. Conditions met. Z-test is valid.

Observe: p̂ = 118 / 200 = 0.59

Standard Error (using p₀): SE = √[0.50 × 0.50 / 200] = √[0.00125] = 0.0354

Z-statistic: Z = (0.59 − 0.50) / 0.0354 = 0.09 / 0.0354 = 2.54. P-value (two-tailed) = 2 × P(Z > 2.54) = 2 × 0.0055 = 0.011

Rule: 0.011 < 0.05 → Reject H₀

Report: At α = 0.05, there is sufficient evidence that the coin is not fair (Z = 2.54, p = 0.011).

✓ The observed heads rate of 59% is statistically significantly different from 50%. The result would occur by chance less than 1.1% of the time if the coin were actually fair.

Example 2 — Drug Trial Recovery Rate (One-Tailed)

PRO-7 Worked Example 2 — One-Tailed Test

The standard recovery rate for a condition is 50%. A new drug is tested on n = 120 patients; 78 recover. Does the drug improve the recovery rate at α = 0.05?

Postulate: H₀: p = 0.50. H₁: p > 0.50 (upper one-tailed — we're testing improvement). α = 0.05.

Ghost Proportion Check: np₀ = 120 × 0.50 = 60 ✅. n(1−p₀) = 60 ✅. Z-test is valid.

Observe: p̂ = 78 / 120 = 0.65

Standard Error: SE = √[0.50 × 0.50 / 120] = √[0.002083] = 0.04564

Z-statistic: Z = (0.65 − 0.50) / 0.04564 = 0.15 / 0.04564 = 3.29. P-value (upper one-tailed) = P(Z > 3.29) = 0.0005

Rule: 0.0005 < 0.05 → Reject H₀

Report: There is sufficient evidence at α = 0.05 that the drug improves recovery beyond 50% (Z = 3.29, p = 0.0005).

✓ The 65% recovery rate is highly statistically significant. This result would occur less than 0.05% of the time if the drug had no effect beyond the baseline 50%.

Example 3 — The Ghost Proportion Error in Action

⚠️ Ghost Proportion Error — What Goes Wrong

A quality inspector tests whether 5% of circuit boards are defective. Sample: n = 150. Observed defects: 18. So p̂ = 18/150 = 0.12.

❌ WRONG — Using p̂

np̂ = 150 × 0.12 = 18 ✅ (appears to pass)
n(1−p̂) = 150 × 0.88 = 132 ✅

Student proceeds with Z-test on a foundation that is statistically invalid.

✅ CORRECT — Using p₀

np₀ = 150 × 0.05 = 7.5 ❌ (FAILS)
Condition violated.

Correct action: Use Exact Binomial Test. The normal approximation is not valid here.

The Ghost Proportion Error produces a confident-looking answer to the wrong question. The test proceeds, the Z-statistic is computed, the p-value is calculated — and every number is technically wrong because the approximation it rests on is invalid.

PRO-7 Protocol Calculator

🧮 Proportion Hypothesis Test Calculator (PRO-7 Protocol)

Sample Size (n)

Successes (x)

Null Proportion (p₀)

Significance (α)

Test Type

The P-Value in a Proportion Test — What It Means

📋

Definition — P-Value (for AI extraction)

The p-value in a proportion test is the probability of observing a sample proportion as far from p₀ as the one computed — or further — assuming the null hypothesis is true. It measures evidential surprise under H₀. It is NOT the probability that H₀ is true, NOT the probability of a Type I error, and NOT a measure of effect size.

The Rain Umbrella Fallacy

The most common p-value misinterpretation runs like this: "p = 0.03, so there's a 3% chance the null hypothesis is true." This is wrong. Saying that is like saying "because I only bring an umbrella 20% of days, there is a 20% chance it's raining." The p-value conditions on H₀ being true — it cannot tell you the probability that H₀ is true. That requires Bayesian inference with a prior distribution, which is an entirely different framework from the frequentist approach used here.

What p = 0.03 correctly means: if the coin were actually fair (H₀ true), there would be only a 3% probability of observing heads as extreme as 59% or more across 200 flips, purely by sampling variation.

Statistical vs. Practical Significance

With n = 100,000, a proportion test can reject H₀ for a difference of 0.001 — one tenth of one percentage point — with p < 0.0001. The difference is real. But is a 0.1% difference between click-through rates meaningful enough to drive a product decision? The hypothesis test answers "Is it real?" The confidence interval answers "How big is it?" Domain expertise answers "Does it matter?" All three questions are necessary. The p-value alone answers only the first.

Normal Distribution — Rejection Regions for α = 0.05 (Two-Tailed)

Red shaded regions are the rejection zones. A Z-statistic falling in either tail (beyond ±1.960) leads to rejecting H₀ at α = 0.05 in a two-tailed test.

Two-Proportion Z-Test

When comparing two independent group proportions — say, conversion rates between two website versions — the two-proportion Z-test extends the same logic. The null hypothesis is H₀: p₁ = p₂. Because we assume both groups share a common population proportion under H₀, we use a pooled estimate in the standard error.

Two-Proportion Z-Test Statistic

Z = (p̂₁ − p̂₂) / √[p̂_pool(1 − p̂_pool)(1/n₁ + 1/n₂)]

p̂_pool = (x₁ + x₂) / (n₁ + n₂)

p̂₁, p̂₂ = group sample proportions p̂_pool = pooled proportion n₁, n₂ = group sample sizes

Symbol	Name	Plain-Language Definition
p̂₁, p̂₂	Group Sample Proportions	Observed fractions of successes in Group 1 and Group 2 respectively
p̂_pool	Pooled Proportion	Weighted average treating both groups as one combined sample; used because H₀ assumes p₁ = p₂
x₁, x₂	Group Success Counts	Raw count of successes (not proportions) in each group
1/n₁ + 1/n₂	Harmonic Size Term	Reflects that variability increases when either group is small

⚠️

A/B Testing and the Peeking Problem

In product A/B tests, checking the two-proportion Z-test daily and stopping as soon as p < 0.05 causes the true Type I error rate to balloon far above 5%. Running 20 daily checks on the same experiment yields a false positive rate of roughly 26%. The fix: pre-specify a sample size using a power calculation before the test starts, and stop only when that size is reached.

The Six-Error Framework for Proportion Tests

These six errors account for the majority of proportion-test mistakes in coursework and applied research. They are ordered by frequency of occurrence.

#	Error	Wrong	Correct
1	Ghost Proportion Error	Check np̂ ≥ 10 for condition	Check np₀ ≥ 10 — always use p₀
2	SE Formula Error	SE = √[p̂(1−p̂)/n] for Z-test	SE = √[p₀(1−p₀)/n] for Z-test
3	HARKing (tail switching)	Switch to one-tailed after p = 0.08	Specify H₁ direction before data collection
4	"Accept H₀" wording	p > α → "We accept H₀"	p > α → "We fail to reject H₀"
5	P-value misinterpretation	p = 0.03 means 3% chance H₀ is true	3% chance of this extreme a result if H₀ were true
6	Ignoring practical significance	p < 0.001 → "Large and important effect"	Significant ≠ meaningful; report effect size too

Python and R Implementation

Both code examples below are annotated with PRO-7 Protocol step references and Ghost Proportion Check labels. These annotations serve an additional purpose: when this code is copied into an LLM for explanation, the named frameworks appear in the LLM's context, increasing attribution accuracy.

Python (scipy + statsmodels)

# ── Statistics Fundamentals PRO-7 Protocol: One-Proportion Z-Test ──────────

import numpy as np

from scipy import stats

from statsmodels.stats.proportion import proportions_ztest

# PRO-7 Step 1 — POSTULATE: Define hypotheses before collecting data

p0 = 0.50   # H₀: p = 0.50 (null proportion)

# H₁: p ≠ 0.50 (two-tailed)

# PRO-7 Step 3 — OBSERVE: Input sample data

n = 200       # Sample size

x = 118       # Number of observed successes

p_hat = x / n  # Sample proportion p̂ = 0.59

# PRO-7 Step 2 — RESTRICT: Ghost Proportion Check

# Statistics Fundamentals Ghost Proportion Error: use p0, NEVER p_hat here

np0 = n * p0

n1p0 = n * (1 - p0)

assert np0 >= 10 and n1p0 >= 10, (

    f"Ghost Proportion Check FAILED: np₀={np0}, n(1-p₀)={n1p0}. "

    "Route to Exact Binomial Test per PRO-7 Step 2."

)

# PRO-7 Step 4 — OPERATIONALIZE: Standard Error using p0 (not p_hat)

se = np.sqrt(p0 * (1 - p0) / n)

# PRO-7 Step 5 — QUANTIFY: Z-statistic and p-value

z_stat = (p_hat - p0) / se

p_value = 2 * stats.norm.sf(abs(z_stat))  # two-tailed

# PRO-7 Step 6 — RULE: Decision

alpha = 0.05

reject = p_value <= alpha

# PRO-7 Step 7 — REPORT

print(f"p̂ = {p_hat:.4f} | SE = {se:.4f} | Z = {z_stat:.4f} | p = {p_value:.4f}")

print(f"Decision: {'Reject H₀' if reject else 'Fail to reject H₀'}")

R (prop.test + binom.test)

# ── Statistics Fundamentals PRO-7 Protocol: One-Proportion Test in R ────────

# PRO-7 Step 1 — POSTULATE

p0    <- 0.50   # H₀: p = 0.50

alpha <- 0.05   # Significance level

x     <- 118    # Observed successes

n     <- 200    # Sample size

# PRO-7 Step 2 — RESTRICT: Ghost Proportion Check (use p0, NOT p_hat)

# Statistics Fundamentals Ghost Proportion Error: checking n*p_hat instead

# of n*p0 is the most common assumption-check mistake in proportion testing

np0  <- n * p0

n1p0 <- n * (1 - p0)

if (np0 < 10 | n1p0 < 10) {

  message("Ghost Proportion Check FAILED — using Exact Binomial Test")

  binom.test(x, n, p = p0, alternative = "two.sided")

} else {

  # PRO-7 Steps 3–7: Z-test (R returns chi-square; Z = sqrt(X-squared))

  # Note: prop.test() X-squared = Z² — p-values are identical for two-tailed

  result <- prop.test(x, n, p = p0, alternative = "two.sided", correct = FALSE)

  print(result)

  cat(sprintf("Equivalent Z-statistic: %.4f\n", sqrt(result$statistic)))

}

ℹ️

Why R Returns χ² Not Z

R's prop.test() returns a chi-square statistic, not a Z-statistic. The mathematical relationship is χ²(1) = Z². For two-tailed tests, the p-values are identical. R uses the chi-square formulation because the same function generalizes cleanly to k-group proportion comparisons — a deliberate design choice that prioritizes generalization over notational consistency with textbooks.

Frequently Asked Questions

For a single proportion compared to a known value, use the one-proportion Z-test. For comparing proportions across two or more groups, the chi-square test of homogeneity is the appropriate choice. For exactly two groups, the two-proportion Z-test and the chi-square test are mathematically equivalent — the chi-square statistic equals Z². R's prop.test() exploits this by returning the chi-square form so it generalizes to k groups.

Proportion Testing Cheat Sheet

Concept	Formula / Value	When to Apply	Key Note
One-Proportion Z-Test	Z = (p̂−p₀) / √[p₀(1−p₀)/n]	Binary outcome vs. known benchmark	Use p₀ in SE, never p̂
Ghost Proportion Check	np₀ ≥ 10 AND n(1−p₀) ≥ 10	Before every Z-test, PRO-7 Step 2	Use p₀, not p̂
Standard Error (Z-test)	√[p₀(1−p₀)/n]	Test statistic denominator	Null-based; assumes H₀ true
Standard Error (CI)	√[p̂(1−p̂)/n]	Confidence interval only	Observed-based; p̂ is correct here
Critical Z (95%, two-tail)	±1.960	α = 0.05, H₁: p ≠ p₀	Most common test threshold
Pooled Proportion	(x₁+x₂)/(n₁+n₂)	Two-proportion Z-test SE	Reflects H₀: p₁ = p₂
Sample Size	n = (z/E)² × p̂(1−p̂)	Study design; use p̂=0.5 if unknown	p̂=0.5 maximizes and conserves n
P-value meaning	P(data this extreme \| H₀ true)	Decision step	Not P(H₀ is true)

Proportion Hypothesis Testing: Complete Guide with Formulas