# Paired-sample and two-sample hypothesis tests

## Paired-Sample Hypothesis Tests

• A paired-sample hypothesis test is used when we have two measurements or observations for the same item or individual. It’s often used when measurements are taken at two different times (like before and after a treatment), or under two different conditions.
• The basic idea of paired-sample hypothesis testing is to reduce the problem to a set of one-sample hypothesis problems. This is done by considering the differences between the paired observations.
• Typically, we form the null hypothesis (H0) stating that there is no change or difference due to the specific condition or treatment and an alternative hypothesis (H1) stating that there is a significant change or difference.
• The exact form of the paired-sample t-test statistic depends on the specific null and alternative hypotheses, but in general it measures the size of the observed effect relative to what would be expected by random chance alone.

## Two-Sample Hypothesis Tests

• A two-sample hypothesis test is used when we want to compare two independent populations. For example, we might want to compare the means of two different groups to see if they’re significantly different.
• In a two-sample test, we’re typically interested in the difference between the population means of the groups we’re comparing. The null hypothesis, in this case, typically assumes that the population means are equal.
• The two-sample t-test is a popular choice when the data is normally distributed. But even when this assumption doesn’t hold, the t-test can be quite robust.
• As with the paired-sample test, we calculate a test statistic which measures the size of the observed effect relative to what would be expected by random chance alone.

## Performing the Tests

• When actually performing these tests, ensure any assumptions underlying the test have been met. These might include normality of the data, homogeneity of variance, or independence of observations.
• Always state your null and alternative hypotheses clearly and decide on your significance level (usually 0.05) before starting the test.
• If the p-value of the test is less than your chosen significance level, then you reject the null hypothesis in favour of the alternative. This indicates a statistically significant effect or difference.
• Remember that failure to reject the null hypothesis does not prove the null is true. It just means that you did not find strong enough evidence to support the alternative.