Confidence Intervals: Intro Stats Chapter 7

Why this chapter matters

Confidence intervals report plausible ranges for unknown population values, which is often more informative than a single estimate.

What you will learn

Build one-sample confidence intervals for means and proportions under stated conditions.
Interpret interval meaning correctly in repeated-sampling terms.
Analyze how confidence level and sample size affect margin of error.

Understand the core ideas

A confidence interval reports a plausible range for an unknown population parameter by combining an estimate with a margin of error. The method uses a standard error and a critical value chosen from a reference distribution. Interpretation is often misstated, so precision in language matters. A 95 percent confidence level means that if you repeatedly sampled and rebuilt intervals with the same method, about 95 percent of those intervals would capture the true parameter. It does not mean there is a 95 percent chance that one fixed computed interval contains a fixed parameter.

Interval width depends mainly on sample size, data variability, and chosen confidence level. Larger samples reduce standard error and usually narrow intervals, while higher confidence levels require larger critical values and produce wider intervals. This is a practical tradeoff between precision and long run coverage reliability. Good reporting includes both interval endpoints and context, then avoids certainty language that overstates what sample data can prove.

Key terms

Confidence interval: An estimated range built from sample data to capture a population parameter.
Margin of error: Half-width of the interval, typically critical value times standard error.
Critical value: Reference cutoff from a distribution used to scale uncertainty.
Confidence level: Long run proportion of intervals that would capture the true parameter.

One-sample mean interval

Sample size is 64, sample mean is 40, and known sigma is 16. Build a 95 percent interval.

Compute standard error: 16 / $\sqrt{64}$ = 2.
Use z critical value 1.96 for 95 percent confidence.
Compute margin of error: 1.96 * 2 = 3.92.
Construct interval 40 plus or minus 3.92, giving 36.08 to 43.92.

Result: Estimated 95 percent confidence interval for population mean is 36.08 to 43.92.

A common misconception

Claim: A 95 percent confidence interval means there is a 95 percent chance this interval contains the parameter.

Correction: For frequentist intervals, the parameter is fixed and the method has 95 percent long run capture rate across repeated samples.

Lessons in this chapter

Point estimates and margin of errorSeparate center estimate from uncertainty width.
Intervals for meansConstruct and read t-based intervals when population spread is unknown.
Intervals for proportionsConstruct and interpret one-proportion confidence intervals.
Choosing confidence and sample sizeUse design goals to balance precision and certainty. Read the full guide →

Study task

Design a poll for a binary opinion question with target margin of error 3 percentage points at 95% confidence, and explain your sample size logic.

Chapter checkpoint

What changes first if you increase confidence level from 90% to 99% with the same data?

The critical value increases, so margin of error increases and the interval becomes wider.

Confidence Intervals