Student's t-test

Student's t-test

The Student's t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups. It is commonly used when the sample sizes are small, and the population standard deviations are unknown. The test was developed by William Sealy Gosset, who published under the pseudonym "Student."

Types of t-tests

There are several types of t-tests, each suited for different experimental designs and data structures:

• One-sample t-test: This test compares the mean of a single sample to a known value or population mean.
• Independent two-sample t-test: This test compares the means of two independent groups to determine if they are significantly different from each other.
• Paired sample t-test: This test compares the means of two related groups, such as measurements taken before and after a treatment on the same subjects.

Assumptions

The Student's t-test relies on several assumptions:

• The data are continuous.
• The data follow a normal distribution.
• The variances of the populations are equal (for the independent two-sample t-test).
• The samples are independent (for the independent two-sample t-test).

Calculation

The t-test statistic is calculated using the formula:

\[ t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

where:

• \(\bar{X}_1\) and \(\bar{X}_2\) are the sample means,
• \(s_1^2\) and \(s_2^2\) are the sample variances,
• \(n_1\) and \(n_2\) are the sample sizes.

The calculated t-value is then compared to the critical value from the t-distribution table, based on the desired level of significance (e.g., 0.05) and the degrees of freedom.

Applications

The Student's t-test is widely used in various fields, including:

• Medicine: To compare the effectiveness of treatments.
• Psychology: To test hypotheses about behavioral data.
• Education: To compare test scores between different teaching methods.
• Business: To compare the performance of different strategies or products.

History

The test was introduced by William Sealy Gosset in 1908 while he was working at the Guinness Brewery in Dublin. Due to the company's policy on publishing research, Gosset used the pseudonym "Student."

See also

• Analysis of variance
• Chi-squared test
• Hypothesis testing
• Normal distribution
• Statistical significance