Confidence Intervals

Confidence Intervals

For an Unknown MeanA Confidence Interval for μ, Normal Distribution with unknown mean μ and known variance σ2Note: A 95% confidence interval means that 95% of the interval's construction - centered around x̅ - will include the true value of μ.

Let X~N(μ,σ^2) where μ is unknown and σ^2 is known.95% confidence intervals for the true value of the population mean μ are:Note: The width of the confidence interval gets smaller as n gets bigger x̅-Zαx√σ^2/n≤μ≥x̅+Zαx√σ^2/n

Note:α is the area in the upper tail of the Normal DistributionZα can be found using Table 4The most common confidence interval values are: 99% Z(0.005)=2.575898% Z(0.01)=2.326395% Z(0.025)=1.9690% Z(0.05)=1.6449

For the Difference of 2 MeansLet X˅1~N(μ˅1,σ^2˅1) and independently X˅2~N(μ˅2,σ^2˅2), where both σ^2 are known.Then 100(1-Zα)% confidence intervals for μ˅1-μ˅2 are:(x̅˅1-x̅˅2)-Zαx√σ^2˅1/n˅1+σ^2˅2/n˅2≤μ˅1-μ˅2≥(x̅˅1-x̅˅2)+Zαx√σ^2˅1/n˅1+σ^2˅2/n˅2