[Maths Class Notes] on Student's t-Test Pdf for Exam

In the area of statistics, a student’s t-test is mentioned as a method of testing the theory about the mean of a small sample drawn from a normally distributed population where the standard deviation of the given population is unknown. 

We can define the Student t-test as a method that tells you how significant the differences can be between different groups. A Student t-test is defined as a statistic and this is used to compare the means of two different populations. 

It is a method that is often used in hypothesis testing to find out whether a process or whether a given treatment actually has any effect on the population of interest, or whether or not two populations are different from each other. You wish to know whether the mean petal length of iris flowers differs according to their distinct species. You find two different species of iris flowers growing in a garden and they measure 25 petals of each species. You can test the difference between these two groups with the help of the Student t-test.

Student t Test Introduction

In the year 1908, an Englishman named William Sealy Gosset developed the t-test as well as t distribution. (Gosset worked at the Guinness brewery in Dublin and found which existing statistical techniques using large samples were not useful for the small sample sizes which he encountered in his work).

The t distribution belonging under a family of curves in which the number of degrees of freedom specifies a particular curve. As the sample size (and the degrees of freedom) increases, the t distribution approaches the bell shape of the standard normal distribution. In common, for tests involving the mean of a sample of size greater than 30, then the normal distribution is applied.

What Type of Student t-Test Should I Use?

When choosing a Student t-test, two things need to be kept in mind: whether the groups being compared are coming from a single population or two different populations, and whether you want to test the difference in some specific direction.

One-Sample, Two-Sample, or Paired t-Test?

• If the groups are coming from a single population (e.g. measuring after and before an experimental treatment), perform a paired t-test.

• If the groups are coming from two different populations (e.g. people from two separate cities), perform a two-sample t-test (also known as independent t-test).

• If there is a group being compared against any standard value (e.g. comparing the acidity of any liquid to a neutral pH of 7), perform a one-sample t-test.

Student t-Test Formula

We have already discussed the t-test definition. The formula for the two-sample t-test (a.k.a. the Student’s t-test) is shown below.

In the formula given above, t is equal to the t-value, x₁ and x₂ are the means of the two groups being compared, s₂is the pooled standard error of the two groups, and n₁ and n₂ are the numbers of observations in each of the groups.

A larger t-value denotes the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups.

You can compare your calculated t-value against the values in a critical value chart to determine whether your t-value is greater than what would be expected by chance. If so, you can reject the null hypothesis and you can conclude which two groups are in fact different.

Uses of Student t Test

Student t test Example: Let’s say you are suffering from a cold and you try a naturopathic remedy. Your cold lasts for a couple of days. The next time you have a cold, you buy an over-the-counter pharmaceutical and the cold lasts a week. You survey your friends and ask which of their colds were of shorter duration when they took the homoeopathic remedy. What you really want to know is, are the results repeatable? A t-test will help you by comparing the means of the two groups and let you know the probability of it happening.

In real life, student’s t-tests can be used to compare averages. For example, a drug company may want to test a new cancer drug to find out if it improves life expectancy or not. In an experiment, there’s always a control group (which is basically a group who are given a placebo, or “sugar pill”). The control group may show an average life expectancy of 5 plus years, while the group taking the new drug might have a life expectancy of 6 plus years. It would seem which drug might work. But it could be due to a fluke. To test this, a student’s t-test can be used to find out if the results are repeatable for an entire population.