CI for Proportions: Gauging Likelihoods

Q: Why do we use $ \hat{p} $ inside the square root instead of the true $ p $?

Because $ p $ is the very parameter we are trying to estimate. We use $ \hat{p} $ as a plug-in estimator, which is valid due to the Slutsky theorem.

Q: What happens when $ \hat{p} $ is close to 0 or 1?

The Wald interval performs poorly. In such cases, the Wilson score interval or Agresti-Coull interval is preferred to avoid intervals that exceed the range [0, 1].

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The Formal Theorem

Let

X_1, X_2, \dots, X_n

be a sequence of independent and identically distributed Bernoulli trials with parameter

p

, where

\hat{p} = \frac{1}{n} \sum_{i=1}^{n} X_i

. By the Central Limit Theorem, as

n \to \infty

, the distribution of

\hat{p}

converges to

N\left(p, \frac{p(1-p)}{n}\right)

. For a confidence level of

1 - \alpha

, the Wald confidence interval is given by:

CI = \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Analytical Intuition.

Imagine you are an explorer in a vast, uncharted territory of Bernoulli outcomes—a coin flip where the probability

p

of landing 'heads' is hidden from sight. We cast a net of

n

trials, capturing the empirical proportion

\hat{p}

. Because we cannot see the true center

p

, we build a scaffolding of certainty around our estimate. The term

\sqrt{\hat{p}(1-\hat{p})/n}

acts as our 'uncertainty compass,' shrinking as our sample size

n

grows, effectively pinning down the elusive

p

within a probabilistic cage. We are not saying

p

is moving; rather, we are constructing a bracket that, if we repeated this experiment infinitely, would capture the true value in

1-\alpha

percent of instances. The critical value

z_{\alpha/2}

is the gateway, dictating how wide our net must be to ensure the truth resides inside. We transform the erratic nature of binary events into a steady, reliable margin of error.

CAUTION

Institutional Warning.

Students frequently conflate the standard error $\sqrt{\hat{p}(1-\hat{p})/n}$ with the population standard deviation. Furthermore, they often mistakenly apply the Wald interval when $n\hat{p} < 5$ or $n(1-\hat{p}) < 5$ , ignoring the violation of the normal approximation.

Academic Inquiries.

Why do we use $\hat{p}$ inside the square root instead of the true $p$ ?

Because $p$ is the very parameter we are trying to estimate. We use $\hat{p}$ as a plug-in estimator, which is valid due to the Slutsky theorem.

What happens when $\hat{p}$ is close to 0 or 1?

The Wald interval performs poorly. In such cases, the Wilson score interval or Agresti-Coull interval is preferred to avoid intervals that exceed the range [0, 1].

Standardized References.

Definitive Institutional SourceCasella, G., & Berger, R. L., Statistical Inference

Foundational

Classifying Statistics: Descriptive vs. Inferential

Exploring the cinematic intuition of Classifying Statistics: Descriptive vs. Inferential.

Intermediate

Scales of Measurement: From Nominal to Ratio

Exploring the cinematic intuition of Scales of Measurement: From Nominal to Ratio.

Intermediate

Parametric vs. Non-Parametric: A Strategic Advantage

Exploring the cinematic intuition of Parametric vs. Non-Parametric: A Strategic Advantage.

Foundational

Probability Fundamentals: The Language of Chance

Exploring the cinematic intuition of Probability Fundamentals: The Language of Chance.

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). CI for Proportions: Gauging Likelihoods: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/statistical-inference-i/ci-for-proportions--gauging-likelihoods

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