Interpreting Confidence Intervals

Correct Interpretation

A 95% confidence interval of $(0.42, 0.58)$ for a proportion means:

✅ "We are 95% confident that the true population proportion is between 0.42 and 0.58."

Incorrect Interpretations

❌ "There is a 95% probability that $p$ is between 0.42 and 0.58."

• The parameter $p$ is a fixed number. It either is or isn't in the interval.

❌ "95% of the data falls between 0.42 and 0.58."

• Confidence intervals are about parameters, not individual data values.

❌ "If we sample again, there's a 95% chance the new $\hat{p}$ will be in this interval."

• The CI is about where $p$ is, not where future $\hat{p}$'s will fall.

What "95% Confidence" Really Means

If we were to take many samples and construct a 95% CI from each one, approximately 95% of those intervals would contain the true parameter.

This is a statement about the method, not about any single interval.

Factors That Affect CI Width

1. Confidence Level

• Higher confidence → wider interval
• Lower confidence → narrower interval
• Trade-off: more confidence = less precision

2. Sample Size

• Larger $n$ → narrower interval (more precision)
• Width decreases proportionally to $\frac{1}{\sqrt{n}}$
• To halve the width, quadruple $n$

3. Variability

• More variability in data → wider interval
• Less variability → narrower interval

Margin of Error

$\text{CI} = \text{point estimate} \pm \text{margin of error}$

The margin of error captures the maximum likely estimation error at the given confidence level.

Confidence Interval and Hypothesis Test Connection

A 95% CI contains all values of the parameter that would not be rejected by a two-sided hypothesis test at $\alpha = 0.05$.

If a hypothesized value falls:

• Inside the CI → fail to reject $H_0$
• Outside the CI → reject $H_0$

AP Tip: The AP exam specifically tests whether you can distinguish between correct and incorrect interpretations. Memorize the correct phrasing and practice using it in context.