The *t*-test was developed by William Sealy Gosset, an English statistician and a beer brewer.

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William Sealy Gosset used this test to produce consistent and also high-quality beer. He published this test in a paper under the pen-name "Student". Hence, this test is also called the Student's

*t*-test.

**Three types of**

*t*-test:**A) One-sample t-test**

Suppose you represent a testing agency. The government wants to know whether the average weight of a certain species of animal is 10 kg. To test this, you take a random sample (of let us say 10 animals) and use the one-sample t-test to check whether the sample mean is 10 kg or is it statistically different from 10 kg.

**B) Two-sample t-test**

You own a shop. There are two major consumer groups: i) those who visit your shop by car and ii) those on a motorbike. You want to test - *is there any difference in spending between these two consumer groups?* For this purpose, you use a two-sample *t*-test.

*t-*tests, which is cumbersome.

*t*-test.

**C) Paired samples**

*t*-testYou are a trainer. You trained 10 students. You want to know if there was any improvement in students' knowledge level due to training. To compare, pre-training scores with post-training scores, you use paired samples *t*-test.

This test may appear similar to a two-sample *t*-test but there is a vital difference. **In paired samples t-test, subjects **(in this example, students)

**are the same.**In paired samples

*t*-test, we compare

**the same subject**(students in this example) using some measures (e.g. scores in exams)

**before and after**the

**interventions**(i.e. training in this example).

**Assumptions of**

*t*-test- Normally distributed data
- If this assumption is not valid, then we have to use non-parametric tests
- Samples are drawn randomly from the population
- Homogeneity of variance (variability of data in each group is similar)
- If this assumption is not fulfilled, then we can use Welch's t-test, also called the unequal variances
*t*-test.