What is Student’s t-Test?
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.
The null hypothesis (H0) is one that tells the true difference between these groups.
The alternate hypothesis (Ha) is one that tells the true difference is different from zero.
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
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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.
FAQs on Student's t-Test
1. What is the primary purpose of a Student's t-test in statistics?
The primary purpose of a Student's t-test is to perform hypothesis testing. It is a statistical method used to determine if there is a significant difference between the means of one or two groups. In essence, it helps you decide whether an observed difference (e.g., between a sample average and a known value, or between the averages of two different groups) is real and statistically meaningful, or if it likely occurred by random chance.
2. What are the main types of Student's t-tests and their uses?
There are three main types of Student's t-tests, each designed for a specific scenario:
One-Sample t-Test: This is used to compare the mean of a single sample to a known or hypothesised population mean. For example, testing if the average height of students in a particular school is different from the national average.
Independent Two-Sample t-Test: This compares the means of two separate, independent groups to determine if they are different from each other. For example, comparing the average test scores of two different classes taught by different teachers.
Paired Sample t-Test: This is used to compare the means of the same group at two different points in time (e.g., before and after a treatment) or on two different conditions. For example, measuring the blood pressure of the same set of patients before and after taking a new medication.
3. What is the general formula for calculating the t-statistic?
The formula for a Student's t-test varies slightly depending on the type. The most fundamental is the one-sample t-test formula, which calculates the t-statistic as:
t = (x̄ - μ) / (s / √n)
Where:
- x̄ (x-bar) is the sample mean (the average of your collected data).
- μ (mu) is the known or hypothesised population mean.
- s is the sample standard deviation, which measures the amount of variation or dispersion in your sample data.
- n is the sample size (the number of observations in your sample).
4. How does a Student's t-test differ from a z-test?
The key difference between a t-test and a z-test lies in the conditions under which they are used, primarily related to sample size and knowledge of the population standard deviation.
A t-test is used when the sample size is small (typically n < 30) and the population standard deviation is unknown. It uses the sample standard deviation as an estimate.
A z-test is used when the sample size is large (typically n > 30) and the population standard deviation is known. With a large sample, the sample's characteristics are considered a better estimate of the population's.
5. What are the key assumptions that must be met to use a t-test correctly?
For the results of a t-test to be valid, several assumptions about the data must be met. The main ones are:
Scale of Measurement: The data must be continuous or ordinal (i.e., numerical).
Random Sampling: The data should be collected from a representative, randomly selected portion of the total population.
Normality: The data in each group should be approximately normally distributed. However, the t-test is robust to violations of this assumption if the sample size is large enough.
Homogeneity of Variance: For a two-sample t-test, the variances of the two populations being compared should be roughly equal.
Violating these assumptions can lead to inaccurate conclusions.
6. How do you interpret the results of a t-test using the p-value?
Interpreting a t-test result involves looking at the p-value it generates. The p-value is the probability of observing your data (or something more extreme) if the null hypothesis (which states there is no difference between the means) were true. The general rule is:
If p-value ≤ 0.05 (or your chosen significance level, α), you reject the null hypothesis. This means the observed difference between the means is statistically significant.
If p-value > 0.05, you fail to reject the null hypothesis. This suggests that the observed difference is not statistically significant and could have happened due to random chance.
7. Can you provide a simple real-world example of using a t-test?
Certainly. Imagine a pharmaceutical company develops a new pill to lower cholesterol. To test its effectiveness, they could use a paired-sample t-test. They would measure the cholesterol levels of 25 patients, have them take the pill for a month, and then measure their cholesterol levels again. The t-test would compare the mean cholesterol level 'before' taking the pill to the mean level 'after'. If the p-value is significant (e.g., p < 0.05), the company can conclude that the pill has a statistically significant effect on lowering cholesterol.
8. Why is it called the 'Student's' t-test?
The name 'Student's t-test' has an interesting origin. It was developed by William Sealy Gosset, a chemist and statistician working for the Guinness brewery in Dublin, Ireland, in the early 20th century. Gosset needed a way to make statistical inferences from small samples, which was common in quality control for brewing. Due to a company policy that prevented employees from publishing research under their own names to protect trade secrets, Gosset published his paper on the test in 1908 under the pseudonym 'Student'. The name has stuck ever since.
