Confidence Intervals

ℹ️ Why It Matters

Point estimates hide how uncertain that estimate is. A confidence interval provides a range of plausible values for an unknown population parameter, along with a measure of how confident we are that the interval captures the true value. Without CIs, every estimate looks equally trustworthy, whether it comes from 5 samples or 5,000. Confidence intervals are the backbone of scientific reporting, clinical trials, political polling, and A/B testing. Every time you read "the effect was 3.2% with a 95% CI of [1.1%, 5.3%]", you are seeing a confidence interval at work.

Overview

A confidence interval is a range $[L, U]$ computed from sample data such that, before observing data, $P(L \leq \theta \leq U) = 1 - \alpha$ . After observing data, we compute specific bounds and say we are $100(1-\alpha)\%$ confident the interval contains $\theta$ . The general structure is point estimate ± critical value × standard error. For means with known $\sigma$ , use the z-interval. When $\sigma$ is unknown (almost always), use the t-interval which has heavier tails reflecting the additional uncertainty from estimating $\sigma$ . For proportions, the Wilson score interval is preferred over the basic Wald interval, especially when $\hat{p}$ is near 0 or 1. Bootstrap CIs make no distributional assumptions and work for any statistic.

Key Concepts

General CI Structure

\text{Point Estimate} \pm \text{Critical Value} \times \text{Standard Error}

Here,

$\text{Point Estimate}$ =Best guess for the parameter (e.g., sample mean)
$\text{Critical Value}$ =z-score or t-score for the desired confidence level
$\text{Standard Error}$ =Estimated standard deviation of the point estimate

Z-Interval (Known σ)

\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Here,

$\bar{x}$ =Sample mean (point estimate of μ)
$z_{\alpha/2}$ =Critical value from standard Normal distribution
$\sigma$ =Known population standard deviation
$n$ =Sample size

T-Interval (Unknown σ)

\bar{x} \pm t_{\alpha/2,\; n-1} \cdot \frac{s}{\sqrt{n}}

Here,

$t_{\alpha/2,\; n-1}$ =Critical t-value with n-1 degrees of freedom
$s$ =Sample standard deviation (estimates σ)

CI for Proportion

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

Here,

$\hat{p}$ =Sample proportion
$n$ =Sample size

Margin of Error

ME = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Here,

$ME$ =Half-width of the CI; maximum expected difference

Sample Size for Desired ME

n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2

Here,

$E$ =Desired margin of error

Common Critical Values

Confidence Level	$\alpha$	$z_{\alpha/2}$	$t_{\alpha/2, 30}$
90%	0.10	1.645	1.697
95%	0.05	1.960	2.042
99%	0.01	2.576	2.750

Factors Affecting CI Width

Factor	Effect on Width	Reason
Higher confidence level	Wider	Must cover more of the sampling distribution
Larger sample size	Narrower	Standard error decreases as $1/\sqrt{n}$
Larger variability	Wider	More uncertainty about the parameter

Quick Example

📝T-Interval: Clinical Trial

A drug trial measures blood pressure reduction in 25 patients: $\bar{x} = 8.2$ mmHg, $s = 3.1$ mmHg. 95% CI with $df = 24$ , $t_{0.025, 24} = 2.064$ :

SE = \frac{3.1}{\sqrt{25}} = 0.62, \quad ME = 2.064 \times 0.62 = 1.28

CI = 8.2 \pm 1.28 = [6.92, 9.48]

Since the entire interval is positive, the drug appears effective. If the interval included zero, we could not conclude the drug works.

📝Election Poll CI

Poll of 1,200 voters: 648 support Candidate A. $\hat{p} = 0.54$ .

SE = \sqrt{\frac{0.54 \times 0.46}{1200}} = 0.0144, \quad ME = 1.96 \times 0.0144 = 0.0282

CI = 0.54 \pm 0.028 = [0.512, 0.568]

The interval excludes 50%, so the lead is statistically significant at the 95% level.

Key Takeaways

📋Summary: Confidence Intervals

Structure: Point estimate ± critical value × SE. Narrower CIs = more precise estimates.
Z vs T: Use z when $\sigma$ is known; use t when $\sigma$ is unknown (almost always). The t-distribution has heavier tails for small $n$ .
Wilson > Wald for proportions: Use Wilson score interval when $\hat{p}$ is near 0 or 1 or $n$ is small.
To halve ME, quadruple $n$ : Since $ME \propto 1/\sqrt{n}$ , precision improvement is expensive.
Bootstrap CIs: Resample with replacement, take percentiles. Works for any statistic without distributional assumptions.
Decision Rule: CI excludes null value → reject $H_0$ at the corresponding $\alpha$ .
Misinterpretation: A 95% CI does NOT mean "95% probability the parameter is in this interval." It means 95% of such intervals would capture the parameter in repeated sampling.
Bayesian Alternative: Credible intervals DO have the interpretation "95% probability the parameter is in this interval."

Deep Dive

For detailed explanations, worked examples, and Python implementations, explore the dedicated statistics lessons: