2-sample problems - comparison of 2 samples, and making inferences of corresponding populations

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

CI with 2 Independent Samples - both 𝑋 and 𝑌 samples come from 2 different subjects (i.e. independent observations)

sample size of 𝑋 and 𝑌 may be different

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 For 2 Independent Samples of Sample Mean

the general formula states the confidence interval is:

• 𝜃ˆ ± 𝑧*·𝑆𝐸(𝜃ˆ)

computing CI for population mean, we substitute:

• 𝜃ˆ = 𝑋̅-𝑌̅
• 𝑆𝐸(𝜃ˆ) = 𝑆𝐸(𝑋̅-𝑌̅) = 𝑟𝑜𝑜𝑡[(𝜎_𝑋²/𝑛_𝑋) + (𝜎_𝑌²/𝑛_𝑌)]

computation of 𝑆𝐸(𝑋̅-𝑌̅):

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• 𝑉𝑎𝑟(𝑋̅-𝑌̅) = 𝑉𝑎𝑟(𝑋̅) + 𝑉𝑎𝑟(𝑌̅) = (𝜎_𝑋²/𝑛_𝑋) + (𝜎_𝑌²/𝑛_𝑌)
• 𝑆𝑡𝑑(𝑋̅-𝑌̅) = 𝑆𝐸(𝑋̅-𝑌̅) = 𝑟𝑜𝑜𝑡((𝜎_𝑋²/𝑛_𝑋) + (𝜎_𝑌²/𝑛_𝑌))

if population standard deviation 𝜎_𝑋 and 𝜎_𝑌are UNKNOWN and assumed to be:

• EQUAL (use pooled standard deviation)

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  how do we estimate 𝜎? we could use a pooled standard deviation (𝑠𝑝):

  • 𝑠_𝑝= [ (𝑛_𝑋 - 1) 𝑠_𝑋² + (𝑛_𝑌 - 1) 𝑠_𝑌²] / [ 𝑛_𝑋 + 𝑛_𝑌 - 2 ]

  therefore:

  • 𝑆𝐸ˆ(𝑋̅-𝑌̅) = 𝑠_𝑝·𝑟𝑜𝑜𝑡(1/𝑛_𝑋 + 1/𝑛_𝑌)

  the 100(1-𝛼)% CI for (𝜇_𝑋-𝜇_𝑌) when :

  • (𝑋̅-𝑌̅) ± 𝑡_{𝛼/2,𝑛𝑋+𝑛𝑌-2}·𝑆𝐸ˆ(𝑋̅-𝑌̅)

  the 100(1-𝛼)% CI for (𝜇_𝑋-𝜇_𝑌) when either (sample sizes of 𝑛_𝑋and 𝑛_𝑌are large) or (𝑋̅ and 𝑌̅ are sampled from a population that has normal distribution):

  • (𝑋̅-𝑌̅) ± 𝑧_𝛼/2·𝑆𝐸ˆ(𝑋̅-𝑌̅)

• NOT EQUAL (use Satterthwaite’s Approximation)

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  Satterthwaite’s Approximation

  • 𝑣 = [𝑠_𝑋²/𝑛_𝑋 + 𝑠_𝑌²/𝑛_𝑌]²/ [ 𝑠_𝑋⁴/(𝑛_𝑋²(𝑛_𝑋-1)) + 𝑠_𝑌⁴/(𝑛_𝑌²(𝑛_𝑌 -1)) ]

  the 100(1-𝛼)% CI for (𝜇_𝑋-𝜇_𝑌) when

  • (𝑋̅-𝑌̅) ± 𝑡_𝛼/2,𝑣·𝑟𝑜𝑜𝑡(𝑠_𝑋²/𝑛_𝑋 + 𝑠_𝑌²/𝑛_𝑌)

CI Formulas For 2 Independent Samples of Sample Mean