Key Takeaways: Confidence Intervals: Estimating with Uncertainty

One-Sentence Summary

A confidence interval combines a point estimate with a margin of error to produce a range of plausible values for a population parameter, where the "95% confidence" describes the long-run reliability of the method (95% of intervals constructed this way would capture the true parameter) — not the probability that any single interval is correct.

Core Concepts at a Glance

The Confidence Interval Architecture

$$\boxed{\text{CI} = \text{Point Estimate} \pm \text{Critical Value} \times \text{Standard Error}}$$

Confidence Interval Formulas

For a Population Mean

$$\boxed{\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}, \qquad df = n - 1}$$

Conditions: 1. Random sample (or random assignment) 2. Independence (10% condition: $n \leq 0.10 \times N$) 3. Nearly normal population or large sample ($n \geq 30$)

For a Population Proportion

$$\boxed{\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}$$

Conditions: 1. Random sample (or random assignment) 2. Independence (10% condition) 3. Success-failure condition: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$

Common Critical Values

For $t^*$, use a t-table or software with $df = n - 1$. Key pattern: $t^* > z^*$ always, but $t^* \to z^*$ as $df \to \infty$.

The t-Distribution vs. Normal

What "95% Confidence" Really Means

The key insight: The parameter is fixed — it doesn't move. The interval is random — it depends on which sample you get. The 95% describes how often the random interval captures the fixed target, across many repetitions of the sampling process.

The Tradeoff Triangle

                  Confidence Level
                       ╱╲
                      ╱  ╲
                     ╱Pick╲
                    ╱ any  ╲
                   ╱  two;  ╲
                  ╱the third ╲
                 ╱is determined╲
                ╱________________╲
   Sample Size ──────────────── Margin of Error

The diminishing returns rule: To halve the margin of error, you must quadruple the sample size. (Same $\sqrt{n}$ relationship as standard error in Chapter 11.)

Sample Size Determination

For a Mean

$$\boxed{n = \left(\frac{z^* \cdot \sigma}{E}\right)^2}$$

For a Proportion

$$\boxed{n = \left(\frac{z^*}{E}\right)^2 \cdot \hat{p}(1-\hat{p})}$$

Conservative approach: Use $\hat{p} = 0.5$ when you have no prior estimate (maximizes the required $n$, guaranteeing the desired MOE).

Always round up to the next whole number.

Python Quick Reference

import numpy as np
from scipy import stats

# --- CI for a Mean ---
x_bar, s, n = 128.3, 18.6, 120

# Method 1: Direct
ci = stats.t.interval(confidence=0.95, df=n-1,
                      loc=x_bar, scale=s/np.sqrt(n))

# Method 2: Manual
t_star = stats.t.ppf(0.975, df=n-1)
moe = t_star * s / np.sqrt(n)
ci = (x_bar - moe, x_bar + moe)

# --- CI for a Proportion ---
x, n = 96, 800
p_hat = x / n
z_star = stats.norm.ppf(0.975)
se = np.sqrt(p_hat * (1 - p_hat) / n)
moe = z_star * se
ci = (p_hat - moe, p_hat + moe)

# --- Sample Size ---
# For a mean:
n_mean = int(np.ceil((1.96 * sigma / E) ** 2))

# For a proportion:
n_prop = int(np.ceil((1.96 / E) ** 2 * p_hat * (1 - p_hat)))

Excel Quick Reference

Common Misconceptions

Worked Example Summary: Maya's Blood Pressure CI

How This Chapter Connects

The One Thing to Remember

If you forget everything else from this chapter, remember this:

A confidence interval is your first inference tool: point estimate ± margin of error. The "95% confidence" means that if you used this method many times, 95% of the resulting intervals would capture the true parameter. The parameter is fixed; the interval is random. The randomness is in which interval you happen to get, not in where the parameter is. To narrow the interval, increase the sample size — but remember the diminishing returns: quadrupling $n$ only halves the margin of error. This tradeoff between precision, confidence, and sample size is the fundamental constraint of statistical estimation, and understanding it makes you a smarter consumer of every poll, study, and data-driven claim you'll ever encounter.

Key Terms