Inference for two samples is a statistical method used to make conclusions about the population parameters based on data collected from two different samples. The goal of this analysis is to determine whether there is a statistically significant difference between the two populations from which the samples were drawn.

There are different types of inference tests that can be used depending on the type of data and the research question being addressed. Some of the most commonly used tests for two-sample inference include:

1. Independent samples t-test: This test is used to compare the means of two independent groups. It assumes that the two samples are normally distributed and have equal variances.
2. Paired samples t-test: This test is used to compare the means of two related groups. It is typically used when the same individuals are measured twice (before and after treatment, for example).
3. Wilcoxon rank-sum test: This non-parametric test is used to compare the medians of two independent groups when the assumption of normality is not met.
4. Mann-Whitney U test: This non-parametric test is similar to the Wilcoxon rank-sum test, but it is used when the two groups have different sample sizes.

These tests are widely used in various fields such as psychology, medicine, engineering, and social sciences to draw conclusions about the differences between two populations.

But note that, the general form of confidence intervals and test statistics will be the same for all of the procedures covered in this  tutorials:

We will be using a five step hypothesis testing procedure again in this lesson:

1. Check assumptions and write hypotheses
The assumptions will vary depending on the test. The null and alternative hypotheses will also be written in terms of population parameters; the null hypothesis will always contain the equality (i.e., = ).

2. Calculate the test statistic
This will vary depending on the test, but it will typically be the difference observed between the sample and population divided by a standard error. In this lesson we will see z and t test statistics.

3. Determine the p- value

4. Make a decision
If p ≤ α reject the null hypothesis. If p > α fail to reject the null hypothesis.

5. State a “real world” conclusion
Based on your decision in step 4, write a conclusion in terms of the original research question.