is a subtype of CI with 2 Samples

2-sample problems - inference on parameters involving two populations

• Population 1: 𝑋 ∼ f_𝑋(𝑥), 𝐄(𝑋) = 𝜇_𝑋
• Population 2: 𝑌 ∼ f_𝑌(𝑦), 𝐄(𝑌) = 𝜇_𝑌

CI with Paired Samples - both 𝑋 and 𝑌 samples come from SAME subject

sample size of 𝑋 and 𝑌 are the SAME

CI - General Formula

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CI Definition

An interval [𝐴, 𝐵] is a (1 − 𝛼)100% confidence interval for the parameter 𝜃 if it contains the parameter with probability (1 − 𝛼):

• 𝐏{𝐴 ≤ 𝜃 ≤ 𝐵} = 1 − 𝛼

where:

• 𝛼 - significance level
• (1 − 𝛼) - confidence level or coverage probability

CI Formula Intuition

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Given a sample of data and a desired confidence level (1 − 𝛼), how can we construct a confidence interval [𝐴, 𝐵] that will satisfy the coverage condition

• 𝐏{𝐴 ≤ 𝜃 ≤ 𝐵} = 1 − 𝛼

first we need to estimate 𝜃

choose an unbiased estimator with normal distribution (e.g. MLE)

use the estimator to take the sample data and estimate 𝜃 point estimate 𝜃ˆ

next we standardize 𝜃ˆ to get a standard normal variable 𝑧:

• 𝑧 = [𝜃ˆ - 𝐄(𝜃ˆ)] / 𝜎(𝜃ˆ)

since 𝜃ˆ was estimated with an unbiased estimator: 𝐄(𝜃ˆ) = 𝜃

• 𝑧 = (𝜃ˆ - 𝜃) / 𝜎(𝜃ˆ)

this variable 𝑧 falls between the standard normal quantiles 𝑞_𝛼/2and 𝑞_1−𝛼/2, denoted by

• -𝑧_𝛼/2= 𝑞_𝛼/2
• 𝑧_𝛼/2= 𝑞_1−𝛼/2

with probability (1 - 𝛼), then:

• 𝐏{-𝑧_𝛼/2 ≤ (𝜃ˆ - 𝜃) / 𝜎(𝜃ˆ) ≤ 𝑧_𝛼/2} = 1 - 𝛼

now rearrange for 𝜃:

• 𝐏{𝜃ˆ - 𝑧_𝛼/2·𝜎(𝜃ˆ) ≤ 𝜃 ≤ 𝜃ˆ + 𝑧_𝛼/2·𝜎(𝜃ˆ)} = 1 - 𝛼

we have obtained two numbers:

• 𝐴 = 𝜃ˆ - 𝑧_𝛼/2·𝜎(𝜃ˆ)
• 𝐵 = 𝜃ˆ + 𝑧_𝛼/2·𝜎(𝜃ˆ)

such that

• 𝐏{𝐴 ≤ 𝜃 ≤ 𝐵} = 1 − 𝛼

CI Formulas

Large Sample Size (𝑛)	Normal Population	𝑆𝐸(𝜃ˆ) Known	Confidence Interval
FALSE	FALSE	EITHER	Bootstrap Method
FALSE	TRUE	FALSE	𝜃ˆ ± 𝑡𝛼/2·𝑆𝐸ˆ(𝜃ˆ)
FALSE	TRUE	TRUE	𝜃ˆ ± 𝑧𝛼/2·𝑆𝐸(𝜃ˆ)
TRUE	EITHER	FALSE	𝜃ˆ ± 𝑧𝛼/2·𝑆𝐸ˆ(𝜃ˆ)
TRUE	EITHER	TRUE	𝜃ˆ ± 𝑧𝛼/2·𝑆𝐸(𝜃ˆ)

where:

• 𝜃ˆ - point estimate/statistic or center of the interval
• 𝑧 - z-score a type of confidence multiplier
• 𝑡 - t-score a type of confidence multiplier
• 𝑆𝐸(𝜃ˆ) or 𝜎(𝜃ˆ) or 𝑆𝑡𝑑(𝜃ˆ) - standard error of the point estimator/statistic
• 𝑆𝐸ˆ(𝜃ˆ) or 𝑠(𝜃ˆ) or 𝑆𝑡𝑑ˆ(𝜃ˆ) - estimated standard error of the point estimator/statistic

CI Annotated

CI Diagram

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CI - Formula

100(1 - 𝛼)% CI for 𝜇_𝐷assuming {𝐷₁, 𝐷₂, …, 𝐷_n} are normal(𝜇_𝐷, 𝜎_𝐷²) distributed

from the sample difference {𝐷₁, 𝐷₂, …, 𝐷_n} we can compute 𝐷̅ and 𝑠_𝐷²

CI when 𝜎 is KNOWN

• 𝐷̅ ± 𝑧*·(𝜎_𝐷/√𝑛)

CI when 𝜎 is UNKNOWN

• 𝐷̅ ± 𝑡*·(𝑠_𝐷/√𝑛)

where:

• 𝐷̅ is the sample mean of differences (𝑋̅ - 𝑌̅)
• 𝑧* is the z distribution
• 𝑡* is the t distribution
• 𝜎_𝐷is the population standard deviation
• 𝑠 is the sample standard deviation
• 𝑛 is size of sample
• (𝜎_𝐷/√𝑛) is the standard error of 𝐷̅
• (𝑠_𝐷/√𝑛) is the estimated standard error of 𝐷̅

approximate 100(1 - 𝛼)% CI for 𝜇_𝐷if 𝑛 is large

• 𝐷̅ ± 𝑧*·(𝜎_𝐷/√𝑛) ~ 𝐷̅ ± 𝑡*·(𝑠_𝐷/√𝑛) ~ 𝐷̅ ± 𝑧*·(𝑠_𝐷/√𝑛)

CIs for Sample Mean