Confidence Intervals

Confidence Intervals

• For an Unknown Mean
• A Confidence Interval for μ, Normal Distribution with unknown mean μ and known variance σ2
• Note: A 95% confidence interval means that 95% of the interval's construction - centered around x̅ - will include the true value of μ.

Slide 2

• 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

Slide 3

• Note:
• α is the area in the upper tail of the Normal Distribution
• Zα can be found using Table 4
• The most common confidence interval values are:
• 99% Z(0.005)=2.5758
• 98% Z(0.01)=2.3263
• 95% Z(0.025)=1.96
• 90% Z(0.05)=1.6449

Slide 4

• For the Difference of 2 Means
• Let 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