T-tests are helpful tools in statistics. They help us figure out if there’s a big difference between the average values of two groups.

You will find T-tests used a lot in data science, especially when we don’t have a lot of data or when we don’t know the standard variation of the population. There are three main types of T-tests:

1. Independent T-test: This one compares the averages of two different groups. For example, it might be used to see if two different teaching methods work better than the other.

2. Paired T-test: This compares the averages of the same group at different times. For instance, we can look at how well students perform before and after they receive training.

3. One-sample T-test: This checks the average of one group to see if it is different from a known average. An example would be to see if the average height of a class is different from the national average.

Key Assumptions

For T-tests to work well, we need to make sure a few things are true:

• Normality: The data should look like a bell curve, especially when there's less data (fewer than 30 samples).

• Independence: Each observation needs to stand alone; they shouldn’t influence each other.

• Equal variances (for independent T-test): The two groups should have similar spreads in their data. We can check this with something called Levene's test.

Formula

To do a two-sample independent T-test, we use a specific formula to calculate the T statistic:

$$T = \frac{\bar{X_1} - \bar{X_2}}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

Here's what those symbols mean:

• $\bar{X_1}$ and $\bar{X_2}$ are the average values from each sample.

• $s_p$ is the combined standard deviation of both samples.

• $n_1$ and $n_2$ are the sizes of the two samples.

When to Use T-Tests

• Comparing two groups: Use a T-test when you have two independent or matched samples.

• Small samples: They are particularly useful when you have a small amount of data.

• Finding significance: T-tests can show if the differences we see are significant. Usually, we look for a significance level (α) of 0.05.

In short, T-tests are very important in both experiments and observational studies. They are essential tools for data scientists trying to analyze data!