Point vs Interval Estimates
Interactive CI Simulator
See what '95% confidence' really means! Click 'Run Simulation' to generate many random samples and their confidence intervals. The red line shows the true population mean (μ = 100). Watch how about 95% of intervals capture the true mean—some miss! This demonstrates that confidence refers to the method's reliability, not any single interval.
Two Types of Estimation
A point estimate gives a single value as our best guess for a population parameter. An interval estimate provides a range of values that likely contains the true parameter. While the point estimate is precise, it's almost certainly wrong. The interval estimate acknowledges uncertainty by providing bounds.
The Need for Uncertainty
Every sample gives a different point estimate due to sampling variability. If we took 100 samples, we'd get 100 different means. A confidence interval captures this variability by providing a range that, in repeated sampling, would contain the true parameter a specified percentage of the time.
Building a Confidence Interval
A confidence interval has three components: (1) the point estimate (sample mean), (2) a critical value from the t or z distribution that determines width based on confidence level, and (3) the standard error which measures sampling variability. Together: CI = point estimate ± margin of error.
Confidence Level Interpretation
What 95% Means
~95 of 100 intervals would contain the true μ
A 95% confidence level does NOT mean there's a 95% probability that μ is in this particular interval. The true μ is fixed - it either is or isn't in the interval. Instead, 95% refers to the long-run success rate of the procedure: if we constructed many intervals this way, about 95% would capture μ.
Choosing Confidence Level
Higher confidence levels produce wider intervals. 90% CI is narrower but misses μ more often. 99% CI is wider but rarely misses. 95% is the most common choice, balancing precision and reliability. The choice depends on the cost of being wrong versus the need for precision.
What CIs Do NOT Mean
Common misinterpretations: CIs don't give probability statements about a fixed parameter, don't describe where data falls, and don't measure subjective confidence. The correct interpretation is frequentist: in repeated sampling, 95% of intervals constructed this way would contain the true parameter.
Margin of Error
Understanding Margin of Error
Margin of error (ME) is the half-width of a confidence interval. It represents the maximum likely difference between the sample estimate and true parameter. ME depends on: (1) confidence level (via t or z), (2) sample variability (s), and (3) sample size (n). Wider margins mean less precision.
What Affects Margin of Error
Three factors affect margin of error: (1) Larger samples decrease ME because √n is in the denominator, (2) More variable data increases ME because s is in the numerator, (3) Higher confidence levels increase ME because the critical value grows. Only sample size is within researcher control.
Poll Margin of Error
In polling, ME shows the uncertainty in reported percentages. With ME = ±3.1%, a reported 52% really means 48.9% to 55.1%. When two candidates' intervals overlap, the race is 'within the margin of error' - we can't confidently declare a leader.
Sample Size and Precision
The √n Relationship
Margin of error decreases with the square root of sample size. This means diminishing returns: to halve your ME, you need 4× the sample size. Going from n=100 to n=400 halves ME. Going from n=400 to n=1600 halves it again. This relationship drives sample size planning.
Sample Size Calculation
To plan a study, determine the desired margin of error, estimate population variability (from pilot study or literature), and solve for n. The formula n = (z × σ / ME)² gives minimum sample size. Always round up, and add extra for potential dropouts or incomplete data.
Precision vs Resources
Sample size involves trade-offs between precision and resources. Larger samples give narrower intervals but cost more time and money. The optimal n is the smallest sample that provides adequate precision for your specific needs. Over-sampling wastes resources; under-sampling wastes the entire study.
Types of Confidence Intervals
CI for Population Mean
For means, use z-interval when population σ is known (rare in practice). Use t-interval when estimating σ from sample s (common). The t-distribution has heavier tails than z, giving wider intervals that account for uncertainty in estimating σ. As n increases, t approaches z.
CI for Population Proportion
For proportions, the standard error depends on p̂ itself. This Wald interval works well when np̂ ≥ 10 and n(1-p̂) ≥ 10. For small samples or extreme proportions, use Wilson score interval or exact binomial methods for better coverage.
CI for Difference Between Groups
CIs for differences help compare two groups. The interval is centered on the difference between sample statistics. If the interval contains zero, the difference is not statistically significant. If it excludes zero, we have evidence of a real difference. This connects CIs directly to hypothesis testing.
Bootstrap Confidence Intervals
Bootstrap CIs don't require distributional assumptions. Resample your data thousands of times, compute the statistic each time, then use percentiles of the bootstrap distribution as CI bounds. Works for any statistic (medians, correlations, ratios) where formulas don't exist.
Real-World Examples
Apply your knowledge to practical scenarios
Election Night Confidence
A poll of n = 1,200 likely voters shows Candidate A at 48% and Candidate B at 44%, with 8% undecided. The margin of error is ±2.8 percentage points at 95% confidence.
Key Insights
- Overlapping CIs indicate the difference isn't statistically significant
- Margin of error applies to EACH candidate, not the gap between them
- Media often report point estimates without uncertainty—ask about the margin
Clinical Trial Results
A drug trial compares a new medication to placebo. Treatment group (n=150): mean blood pressure reduction 12 mmHg, SD = 8. Placebo group (n=150): mean reduction 4 mmHg, SD = 7. Calculate the 95% CI for the difference.
Sample Data
{
"treatment": {
"n": 150,
"mean": 12,
"sd": 8
},
"placebo": {
"n": 150,
"mean": 4,
"sd": 7
}
}Key Insights
- CI excluding 0 for a difference = statistically significant effect
- Look at the entire interval: even the lower bound should be meaningful
- Clinical significance (Is 6 mmHg useful?) differs from statistical significance
The 95% Misunderstanding
A researcher calculates a 95% CI for the mean height of adult males: [175.2 cm, 178.8 cm]. Their colleague asks: 'So there's a 95% chance the true mean is in this interval?'
Key Insights
- 95% confidence = 95% of such intervals contain μ (procedural interpretation)
- It does NOT mean 95% probability for this specific interval
- Bayesian credible intervals allow the probability interpretation people want
Confidence Intervals Quiz
Test your understanding • Passing score: 70%