Zintervallet

Zintervallet, or z-interval, is a confidence interval for a population parameter constructed using the standard normal distribution. It is primarily used to estimate a population mean when the population standard deviation σ is known or when the sample size is large enough for the normal approximation to be reliable.

For estimating μ with known σ, the z-interval is x̄ ± zα/2 · (σ/√n), where x̄ is the sample mean, n the sample size, and zα/2 the critical value from the standard normal distribution for the chosen confidence level (for example, 1.96 for 95% confidence).

If σ is unknown and the sample size is small, a t-interval is preferred; for larger samples, s can substitute for σ and the z-critical value is often used as an approximation. For very large samples the z-interval remains a good approximation.

The z-interval can also be used for proportions: p̂ ± zα/2 √(p̂(1−p̂)/n), where p̂ is the sample proportion.

Assumptions include independent observations and correct specification of σ or the use of a sufficiently large n. Limitations: sensitive to the assumption of a normal model or accurate variance estimate. Compared with the t-interval, the z-interval tends to be narrower when σ is known and n is large, but can be biased if the assumptions are violated.

Example: x̄ = 100, σ = 15, n = 25, 95% level. Margin of error = 1.96 × 15 / √25 = 5.88, yielding [94.12, 105.88].