19TwoSampleTTests

# 19TwoSampleTTests - Statistical Techniques I EXST7005...

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Statistical Techniques I EXST7005 Two-sample t-tests

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Two-sample t-tests Recall the derived population of sample means. Imagine you had two of these derived populations, as we had for variances when we developed the F-test.
Equality of Population Means Given two populations, test for equality. Actually test to see if the difference between two means is equal to some value, usually zero. H0: μ 1 - μ 2 = δ where often δ = 0 Pop'n 1 Pop'n 2 Mean μ 1 μ 2 Variance σ 21 σ 22

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Derived populations From population 1 draw all possible sample of size n1 to get f8e5 Y1. Mean μ 1 yields μ 1 = μf8e5 Y1 Variance σ 21 yields σ 2 f8e5 Y1 = σ 21 / n1 Derived population size is N1n1 Likewise population 2 gives f8e5 Y2. Mean μ 2 yields μ 2 = μf8e5 Y 2 Variance σ 22 yields σ 2 f8e5 Y2 = σ 22 / n 2 Derived population size is N 2 n2
Derived populations (continued) Draw a sample from each population Pop'n 1 Pop'n 2 Sample size n1 n2 d.f. γ 1 γ 2 Sample mean f8e5 Y1 f8e5 Y2 Sample variance S21 S22

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Derived populations (continued) Take all possible differences ( f8e5 Y1- f8e5 Y2)= f8e5 dk i. e. Take all possible means from the first population (N1n1 means) and all possible means from the second population (N 2 n2), an get the difference between each pair. There ar a total of (N1n1)(N 2 n2) values. dk = ( f8e5 Y1i- f8e5 Y2j) for i=1,. ..,N1n1 j=1,. ..,N 2 n2 k = 1,. ..,(N1n1)(N 2 n2)
Derived populations (continued) The population of differences ( f8e5 Y1- f8e5 Y2)= f8e5 dk has mean equal to μf8e5 d or μf8e5 Y1- f8e5 Y2 has variance equal to σ 2 f8e5 d or σ 2 f8e5 Y1- f8e5 Y2 has standard deviation equal to σf8e5 d or σf8e5 Y1- f8e5 Y2

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Characteristics of the derived population 1) As n1 and n2 increase, the distribution of f8e5 d's approaches a normal distribution. If the two original populations are normal, the distribution of f8e5 d is normal regardless of n1 and n2. 2) The mean of the differences ( f8e5 d) is equal the difference between the means of the two parent populations. μf8e5 d = μf8e5 Y1 - μf8e5 Y2 = δ
Characteristics of the derived population (continued) 3) The variance of the differences ( f8e5 d) is equal to the difference between the means o the two parent populations. σ 2 f8e5 d = σ 2 f8e5 Y1- f8e5 Y2 = σ 21 /n1 + σ 22/n2 σf8e5 d = σf8e5 Y1- f8e5 Y2 = SQRT( σ 21 /n1 + σ 22/n2)

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Characteristics of the derived population (continued) The variance comes from linear combinations. The variance of the sums is the sum of the variance (no covariance if independent). Linear combination: f8e5 Y1- f8e5 Y2 Coefficients are 1, -1 Variance is 12*Var( f8e5 Y1) + (-1)2*Var( f8e5 Y2) σ 21 /n1 + σ 22/n2
H0: μ 1 - μ 2 = δ H1: μ 1 - μ 2 δ the non-directional alternative, a one taile test would specify a difference > δ or < δ . Commonly,

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## 19TwoSampleTTests - Statistical Techniques I EXST7005...

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