Chapter 1 Two Sample t-Test

The two-sample t-test is by far the most powerful test when the assumptions are met for comparing means between two samples. Assumptions:

Random sample from each population
Both samples are independent
Both population distributions are normal
Both population variances are equal

Note: By the Central Limit Theorem, we can assume that the sample means will start looking normal at large sample sizes (\(n \geq 40\)).

How It Works

Do we know the population variance \(\sigma^2\)? - If so, we’ll use the \(z\) distribution: \(z\sim N(0,1)\) - Otherwise, we’ll use the \(t\) distribution, \(t\sim t(df)\), where \(df\) is the minimum of the two sample sizes - 1.

Code

library(stats)

sample1 <- c(1.1, 2.1, 4.2, 3.2, 1.7, 2.2, 2.7)
sample2 <- c(3.9, 2.9, 3.8, 1.8, 3.3, 2.8, 2.3)

t.test(x=sample1, y=sample2, alternative='two.sided', paired=F)

from scipi.stats import ttest_ind

sample1 = [1.1, 2.1, 4.2, 3.2, 1.7, 2.2, 2.7]
sample2 = [3.9, 2.9, 3.8, 1.8, 3.3, 2.8, 2.3]

ttest_ind(a=sample1, b=sample2)