t-test

Uses

• characterize the relationship between two samples to test the hypothesis that they have been sampled from the same underlying population

Parametric methods for the comparison of sample means

• Student's t-Test

How this is done

Consider the situation where you wish to compare 2 samples, each containing n values. You are hoping to statistically evaluate whether these samples could have been derived from the same underlying distribution or whether this scenario is unlikely. You specifically test the H_o: µ₁ = µ₂. To test the null hypothesis that the two sample means are derived from the same population, we will place the difference between the two means within the context of the population's standard deviation. Depending on the samples N these means should be found within a certain confidence interval.

Step-by-step: Note that when you collect small data sets from the same underlying, normal distribution, the means of these samples will all vary slightly due to chance differences in the actual values sampled. Also variances from different samples will differ from each other due to chance alone. As data points are normally distributed around their sample means with a given variance s²=S(Y_i-)/n-1, so the sample means will be normally distributed around a mean of means with a given standard error SE = ^s/

Assumptions

• Independence of datapoints
• Normality and <Central Limit Theorem> Distribution of sample means approaches a normal probability distribution as sample size increases, regardless of the shape of the population from which items are sampled. A sample size of 30 is often regarded as sufficient to employ the central limit theorem
• Homoscedasticity: Homogeneity of variances