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Confidence Intervals: Summary of Formulas

Module by: Dr. Barbara Illowsky, Susan Dean

formula 1: General form of a confidence interval

( lower value , upper value ) = ( point estimate - error bound , point estimate + error bound ) (lower value,upper value)=(point estimate-error bound,point estimate+error bound)

formula 2: To find the error bound when you know the confidence interval

error bound = upper value - point estimate error bound=upper value-point estimate OR error bound = upper value - lower value 2 error bound= upper value - lower value 2

formula 3: Single Population Mean, Known Standard Deviation, Normal Distribution

Use the Normal Distribution for Means EBM = z α 2 ⋅ σ n EBM= z α 2 ⋅ σ n

The confidence interval has the format ( x¯ - EBM , x¯ + EBM ) ( x -EBM, x +EBM).

formula 4: Single Population Mean, Unknown Standard Deviation, Student-t Distribution

Use the Student-t Distribution with degrees of freedom df = n - 1 df=n-1. EBM = t α 2 ⋅ s n EBM= t α 2 ⋅ s n

formula 5: Single Population Proportion, Normal Distribution

Use the Normal Distribution for a single population proportion p ' = x n p'= x n

EBP = z α 2 ⋅ p ' ⋅ q ' n p ' + q ' = 1 EBP= z α 2 ⋅ p ' ⋅ q ' n p'+q'=1

The confidence interval has the format ( p ' - EBP , p ' + EBP ) (p'-EBP,p'+EBP).

formula 6: Point Estimates

x¯ x is a point estimate for μ μ

p ' p' is a point estimate for ρ ρ

s s is a point estimate for σ σ

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