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6.2.6 Two Sample t-test and Confidence Interval

Assumptions: Observations are independent of one another, if the data comes from a non-normal distribution, n1 > 30 and n2 > 30, and the data come from two independent samples. If the data come from a normal distribution, n1 and n2 can be any size for each sample.

H0 : μ1 = μ2

In the case where the variances are not assumed to be equal,

¯x1 - ¯x2 t = ∘-s2---s2. n11 + n22

A (1 - α)100% confidence interval for μ is,

( ∘ --------- ∘ ---------) 2 2 2 2 ( ¯x1 - ¯x2 - tα∕2,r -s1+ s2-,x¯1 - x¯2 + tα∕2,r s1-+ s2-) , n1 n2 n1 n2

where r equals

[ ]2 s21+ s22 r = ----n1---n2-----. (s21∕n1)2-+ (s22∕n2)2 n1-1 n2-1

The value of r is often not an integer, and in that case the nearest integer rounded down is often used. In the case where the variances are assumed to be equal,

¯x1 - ¯x2 t = --∘--1----1. sp n1 + n2

A (1 - α)100% confidence interval for μ is,

( ∘ --------- ∘ --------) -1- 1-- 1-- -1- ¯x1 - ¯x2 - tα∕2,(n1+n2-2)sp n1 + n2, ¯x1 - ¯x2 + tα∕2,(n1+n2-2)sp n1 + n2

where,

∘ ---------2-----------2- (n1---1)s1 +-(n2---1-)s2- sp = n1 + n2 - 2 .

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