Summary Concept 3 and 4 - Two Sample Hypothesis and Paired-sample Hypothesis

Two –Sample Hypothesis
Comparing two populations




Comparing two population means

Are the 2 samples independent?


No, paired Yes



Paired sample t-test Do the two populations have
equal variances?


Yes No


Two-sample t-test for Two-sample t-test with
Homoscedastic equal variances unequal variances
Heteroscedastic (standard method) (Behrens-Fisher test)

, Homoscedastic – Two sample t-test for equal variances

Assumptions:
• The two samples (experimental units) were selected randomly
• The two sets of treatment levels are independent
• The population follows a normal distribution
• The two populations have equal variances


̅𝟏 − 𝑿
We are testing for difference between means 𝑿 ̅𝟐


(𝑋̅1 − 𝑋̅2 ) − 𝜇0
𝑡=
𝑠𝑋̅1 −𝑋̅2


…………………………………………………………………………………………….


Standard error of 𝑋̅1 − 𝑋̅2 :




Because we are assuming equal variances we get a more accurate estimate
by pooling the two groups into a common variance 𝑠𝑝2 :




If we have a balanced design: