Confidence Interval Lower Limit
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A confidence interval is the sample mean flanked by a margin of error, and this is its lower edge: back off from x̄ by z standard errors. A sample of 100 parts averaging 50.0 mm with σ = 10 mm gives a 95% lower limit of 50.0 − 1.96 × (10/10) = 48.04 mm. Jerzy Neyman set the logic out in 1937, and he was careful about what it claims: the procedure captures the true mean 95% of the time over many repetitions. Any one interval either contains μ or it does not.
That is the trap almost everyone falls into — reading "95% confidence" as "there is a 95% chance μ lies in here". The randomness lives in the interval, not in μ. In practice the lower limit is often the number that matters commercially: if a supplier guarantees a mean tensile strength above 48 MPa and your 95% lower limit lands at 48.04, you have just barely cleared the bar. Run the formula backwards to size a study — to keep the lower limit within 4 units of a mean of 100 when σ = 20 at z = 2, you need n = (2 × 20/4)² = 100.
- = Lower confidence limit
- = Sample mean
- = Critical z-value
- = Standard deviation
- = Sample size
- Lower confidence limit — Confidence Interval Upper Limit, Outlier Lower Fence
- Sample mean — Confidence Interval Upper Limit, One-Sample Z-Test Statistic
- Critical z-value — Margin of Error for a Mean, Confidence Interval Upper Limit
- Standard deviation — Z-Score (Standard Score), Variance and Standard Deviation
- Sample size — Standard Error of the Mean, Margin of Error for a Mean