Test of Significance: t-Test

Introduction 

            The t-test is a statistical method used to determine if there is a significant difference between the means of two groups. Its history dates back to the early 20th century when William Sealy Gosset, an English statistician working for the Guinness brewery in Dublin, Ireland, developed the test. Gosset faced a challenge in his work at Guinness because the brewery's small sample sizes made it difficult to use traditional statistical methods. To address this issue, Gosset derived what is now known as the t-distribution and developed the t-test as a way to compare the means of small samples.

            In 1908, Gosset published his work under the pseudonym "Student" in the scientific journal Biometrika. This is why the t-test is sometimes referred to as Student's t-test.

            The t-test has become one of the most widely used statistical tests in scientific research. It has been adapted and expanded upon in various ways, leading to the development of different versions of the test, such as the independent samples t-test, paired samples t-test, and one-sample t-test, each suited to different research scenarios.

Steps to calculate t-test: -

1. Ensure that the data meet the assumptions of the t-test, such as being normally distributed and having similar variances between groups.

2. Date on minimum two groups are required for t-test.

3. State the null hypothesis (H0) and alternative hypothesis (H1)

4. Calculate the means of each group.

5. Calculate the standard deviation for each group.

6. Calculate the standard error of each group: The standard error is the standard deviation divided by the square root of the sample size.

7. Add the standard error of both the groups and calculate the square root.

8. Calculate the t-value: The t-value is calculated as the difference between the means of the two groups divided by the value of standard error arrived at by square root.

9. Determine the degrees of freedom: The degrees of freedom for an independent two-sample t-test is calculated as (df = n1 + n2 - 2 \).

10. Find the critical value: Use a t-distribution table or a statistical software to find the critical t-value based on your chosen significance level (e.g., 0.05 for a 95% confidence level) and degrees of freedom.

11. Compare the calculated t-value with the critical value: If the calculated t-value is greater than the critical value, you reject the null hypothesis and conclude that there is a significant difference between the two groups. Otherwise, you fail to reject the null hypothesis.

*********