Definition:

The t-test is a parametric statistical test that compares the means of two or more groups to determine whether there is a significant difference between them. It is commonly used in independent samples t-tests to compare the means of two groups and in paired samples t-tests to compare the means of two groups that are related in some way.

Assumptions:

• Independent samples t-test:
  • The samples are independent of each other.
  • The data is normally distributed within each group.
  • The variances of the groups are equal.
• Paired samples t-test:
  • The samples are paired or related.
  • The data is normally distributed within each group.
  • The variances of the groups are equal.

Types of t-tests:

• Independent samples t-test: Compares the means of two or more groups that are not related to each other.
• Paired samples t-test: Compares the means of two groups that are related to each other.
• One-sample t-test: Tests whether the mean of a group is equal to a specified value.

Formula:

The t-statistic is calculated as:

$$t = frac{bar{x}_1 – bar{x}_2}{s_p sqrt{n_1 n_2}}$$

where:

• $bar{x}_1$ is the sample mean of group 1
• $bar{x}_2$ is the sample mean of group 2
• $s_p$ is the pooled standard deviation
• $n_1$ is the number of participants in group 1
• $n_2$ is the number of participants in group 2

Critical value:

The critical value is the value of the t-statistic that corresponds to a specified probability level (e.g., 0.05). If the t-statistic is greater than the critical value, then the null hypothesis is rejected and it is concluded that there is a significant difference between the means of the groups.

Conclusion:

The t-test is a powerful statistical test for comparing the means of multiple groups. It is widely used in various fields of research to assess differences between groups and to draw inferences about population parameters. However, it is important to note the assumptions underlying the t-test and to be aware of the potential limitations of the test.