Student's t-test
Introduction
The Student's t-test is a statistical hypothesis test that follows a Student's t-distribution under the null hypothesis. It is used to determine whether there is a significant difference between the means of two groups. The test was developed by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland, who published under the pseudonym "Student".
History and Development
Gosset developed the t-test as a way to cheaply monitor the quality of stout. The test was published in Biometrika in 1908, but it was not until 1931, after the work of Fisher, that the test became widely known and used.
Assumptions
The Student's t-test makes several assumptions about the data it is used on:
1. Independence: The data were sampled independently from the two populations being compared. 2. Normality: The data follow a normal distribution. 3. Homogeneity of variance: The variances of the two populations are equal.
Violation of these assumptions can lead to incorrect conclusions.
Types of t-tests
There are three main types of t-tests: Independent samples t-test, Paired sample t-test, and One sample t-test.
Independent samples t-test
The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared.
Paired sample t-test
The paired sample t-test is used when the samples are dependent; this typically occurs when the test is being used to investigate the differences between scores at two different times (for example, before and after a treatment).
One sample t-test
The one sample t-test is used when we want to compare a sample mean with a population mean which we already know.
Calculations
The t-value is calculated using the formula:
t = (X̄1 - X̄2) / √ ((s1^2/n1) + (s2^2/n2))
where X̄1 and X̄2 are the sample means, s1^2 and s2^2 are the sample variances, and n1 and n2 are the sample sizes.
The degrees of freedom for this test are n1 + n2 - 2.
Interpretation
Once the t-value and degrees of freedom are calculated, a p-value can be found using a t-distribution table. This p-value is used to determine the significance of the results. If the p-value is less than the chosen alpha level (typically 0.05), the null hypothesis is rejected and the difference between the groups is considered statistically significant.
Limitations and considerations
While the t-test is a powerful tool, it is not without its limitations. It is sensitive to outliers and the assumption of normality. If these assumptions are violated, the results of the test may not be valid. In such cases, a non-parametric alternative such as the Mann-Whitney U test may be more appropriate.