Master the art of statistical estimation and uncertainty quantification
A COVID vaccine shows 95% effectiveness in trials. But what's the confidence interval? Understanding this determines public health policy affecting millions.
Real case: Pfizer's vaccine had 95% efficacy with CI [90.3%, 97.6%]
A factory must ensure chip defect rates stay below 0.001%. How many chips must they test to be confident? Too few risks recalls; too many wastes resources.
Industry standard: Sample sizes calculated for 99% confidence
An A/B test shows version B converts 2% better. Is this real or just random variation? Estimation techniques prevent costly mistakes.
Rule: Don't act unless CI excludes zero difference
Master estimation, and you'll make better decisions with incomplete information – a superpower in any field.
These four concepts form the foundation of all statistical estimation:
A single value that represents our best guess for an unknown population parameter
Sample mean x̄ = 72.5 estimates population mean μ
A range of values that likely contains the true population parameter
95% CI: [70.2, 74.8] means we're 95% confident μ is in this range
Measures how much our estimate would vary if we repeated the sampling process
SE = 1.2 means sample means typically vary by ±1.2 from the true mean
The ± amount in a confidence interval, representing precision
Poll result: 52% ± 3% means ME = 3 percentage points
Imagine you're a pharmaceutical company testing a new medication. You can't test it on all 8 billion people on Earth – that would be impossible, unethical, and prohibitively expensive. Instead, you test it on perhaps 1,000 carefully selected volunteers. But here's the critical question: How do you use data from those 1,000 people to make confident statements about how the drug will work for everyone?
This is the essence of statistical estimation. We use samples (the few) to estimate population parameters (the many). But unlike a simple guess, statistical estimation gives us two crucial things:
In this chapter, you'll master both types of estimation. You'll learn when to use each method, how to calculate them correctly, and most importantly, how to interpret them properly – a skill that even many professionals struggle with.
Foundation: Understanding Inference
Core concepts of estimation and how samples relate to populations
Known Variance: The Ideal Case
Confidence intervals when population standard deviation is known
Sample Size: Planning Studies
Calculate how many observations you need before collecting data
Unknown Variance: The Real World
Handle uncertainty when you must estimate both mean and variance
Proportions: Beyond Means
Extend your skills to percentages, rates, and categorical data
Estimation builds on concepts from Chapter 4. Let's quickly review the essentials:
The distribution of a statistic (like x̄) across all possible samples of size n
Key insight: Even if individual data varies wildly, sample means cluster predictably around the population mean
Sample means approach a normal distribution as n increases, regardless of the population shape
Why it matters: This allows us to use normal-based methods for estimation in most real-world scenarios
Measures how much sample statistics vary from sample to sample: SE = σ/√n
Critical concept: Smaller SE means more precise estimates. SE decreases as sample size increases
Population: μ (mean), σ (std dev) | Sample: x̄ (mean), s (std dev)
Remember: We use sample statistics (known) to estimate population parameters (unknown)
💡 Not familiar with these concepts? Review Chapter 4 first, especially the sections on sampling distributions and the CLT.
From samples to population truths
All sections are accessible - follow the numbered sequence for the best learning experience
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Build formulas step-by-step
Master estimation formulas by building them interactively. Click on each part to understand why it's there and how it works.
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From samples to populations
Discover how statisticians perform "magic" – using small samples to make accurate statements about entire populations. You'll explore the difference between what we can measure (statistics) and what we want to know (parameters), and learn both classical and Bayesian approaches to inference.
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Quantifying certainty with known variance
Learn to construct confidence intervals when you know the population standard deviation (common in quality control and standardized testing). You'll discover why 95% confidence doesn't mean "95% probability" and master the famous 68-95-99.7 rule that appears everywhere in statistics.
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Master CI problems through practice
Solidify your understanding with multiple choice quizzes, worked examples, and interactive calculators. Practice common exam questions, avoid typical mistakes, and build confidence in solving CI problems.
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How many observations do we need?
One of the most practical questions in statistics: "How large should my sample be?" Too small and your results are unreliable; too large and you waste resources. Explore the mathematical relationships between sample size, precision, and confidence through interactive 3D visualizations that make these trade-offs crystal clear.
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Real-world estimation with t-distribution
In reality, we rarely know the population standard deviation. Enter the t-distribution – wider and more conservative than the normal distribution, especially for small samples. You'll see why "Student" (W.S. Gosset) developed this while working at Guinness brewery, and explore modern bootstrap methods that work without any distribution assumptions.
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From polls to quality control
Proportions are everywhere: election polls, conversion rates, defect rates, and test pass rates. But proportion intervals behave differently than mean intervals, especially near 0% or 100%. Compare the simple (but flawed) Wald method with the superior Wilson method, and see why poll results need larger samples than you might think.
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See the relationship
Explore how confidence intervals and hypothesis tests are connected. This interactive visualization shows why a 95% CI corresponds to a two-sided test at α=0.05.
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What confidence really means
Build intuition for what confidence intervals actually mean. Work through common misconceptions with interactive examples.
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Why 68-95-99.7?
Understand where these percentages come from and how they connect to confidence intervals. Interactive visualization of the normal distribution.
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Start with fundamentals to understand the core concepts
Apply concepts to real-world data to reinforce understanding
Learn when and how to apply each concept effectively