Confidence Intervals for the Mean — z and t Intervals
Foundations of Statistics
Estimating Means with Precision
Confidence intervals provide a range of plausible values for the population mean, offering more information than point estimates alone. Choosing between z and t intervals depends on whether σ is known, a critical distinction in practice.
- Clinical Trials — Estimating treatment effect sizes with quantified uncertainty
- Business Planning — Forecasting average customer lifetime value with confidence bounds
- Policy Analysis — Measuring intervention effects with appropriate uncertainty
Confidence intervals turn point estimates into informed decisions.
Core Concepts
A confidence interval gives a range of plausible values for the population mean , based on sample data. The choice between z-interval and t-interval depends on whether is known.
z-Interval ( Known)
t-Interval ( Unknown)
Factors Affecting Width
Worked Example: z-Interval
A factory produces bolts with known mm. A sample of bolts has mm. Construct a 95% CI for .
Step 1. The critical value: .
Step 2. The standard error: .
Step 3. The margin of error: .
Step 4. The 95% CI: mm.
Interpretation: We are 95% confident that the true mean bolt length is between 12.14 and 12.46 mm.
Worked Example: t-Interval
A pharmacologist measures drug half-life in patients: hours, hours. Construct a 95% CI for .
Step 1. With degrees of freedom: .
Step 2. Standard error: .
Step 3. Margin of error: .
Step 4. The 95% CI: hours.
Comparison: If we had (incorrectly) used the z-interval: . The t-interval is wider, correctly reflecting the additional uncertainty from estimating with from only 9 observations.