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PAIRED DATA

Here we have a single sample set of size n. However, each of the n members of the sample set is a pair of random variables, each from its own population, i.e., we have (X₁,Y₁), (X₂,Y₂),..., (X_n,Y_n) where

E[X_i]=m₁, S.D.[X_i]=s₁ and E[Y_i]=m₂, S.D.[Y_i]=s₂for all i=1,2,...,n.

Once again, we are interested in testing H₀: m₁-m₂ = D₀. However, we now deal with the random variable D_i=X_i-Y_i.

While the D_i’s are independent of each other, X_i and Y_i for a given i need not be independent of each other.
m_D = E[D] = E[X-Y] = E[X]-E[Y] = m₁-m₂.
It is assumed that the D_i’s are Normally distributed
= +- 2Cov(X,Y).

Define

=sample mean and s_D = sample S.D. of the differences.

PAIRED t-Test: To test H₀: m_D = m₁-m₂ = D₀
the Test Statistic is t =

H_a: m_D>D₀Þ Reject if t ³t_a_,n-1
H_a: m_D<D₀Þ Reject if t £-t_a_,n-1
H_a: m_D¹D₀Þ Reject if t³ t_a_/2,n-1 OR t£-t_a_/2,n-1, i.e., |t|³t_a_/2,n-1

A 100* (1-a)% confidence interval for m_D is given by

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Jayant Rajgopal