Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter. Unlike a hypothesis test, which gives a yes/no decision, a confidence interval quantifies the precision of an estimate.

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The Concept

A 95% confidence interval for a parameter $\theta$θ means: if we repeated the sampling process many times, approximately 95% of the intervals constructed would contain the true value of $\theta$θ.

Important: A 95% confidence interval does NOT mean there is a 95% probability that $\theta$θ lies in the interval. The interval is fixed once calculated; $\theta$θ either is or is not in it. The 95% refers to the long-run success rate of the procedure.

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Confidence Interval for the Mean ($\sigma$σ Known)

If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$X1​,…,Xn​∼N(μ,σ2) and $\sigma$σ is known:

$\bar{X} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$Xˉ±zα/2​⋅n​σ​

Confidence level	$z_{\alpha/2}$zα/2​
90%	1.645
95%	1.960
99%	2.576

Worked Example

A machine fills bottles with mean volume $\mu$μ and known standard deviation $\sigma = 5$σ=5 ml. A sample of 25 bottles gives $\bar{x} = 502$xˉ=502. Find a 95% confidence interval for $\mu$μ.