“ T Test (Student’s T-Test)”

Contents:

  1. What is a T Test?
  2. The T Score
  3. T Values and P Values
  4. Calculating the T Test
  5. What is a Paired T Test (Paired Samples T Test)?

What is a T test?

The t test tells you how significant the differences between groups are; In other words it lets you know if those differences (measured in means/averages) could have happened by chance. A very simple example: Let’s say you have a cold and you try a naturopathic remedy. Your cold lasts 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 they all tell you that their colds were of a shorter duration (an average of 3 days) when they took the homeopathic remedy. What you really want to know is, are these results repeatable? A t test can tell you by comparing the means of the two groups and letting you know the probability of those results happening by chance.

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

The T Score.

The t score is a ratio between the difference between two groups and the difference within the groups. The larger the t score, the more difference there is between groups. The smaller the t score, the more similarity there is between groups. A t score of 3 means that the groups are three times as different from each other as they are within each other. When you run a t test, the bigger the t-value, the more likely it is that the results are repeatable.

A large t-score tells you that the groups are different. A small t-score tells you that the groups are similar.

T-Values and P-values

How big is “big enough”? Every t-value has a p-value to go with it. A p-value is the probability that the results from your sample data occurred by chance. P-values are from 0% to 100%. They are usually written as a decimal. For example, a p value of 5% is 0.05. Low p-values are good; They indicate your data did not occur by chance. For example, a p-value of .01 means there is only a 1% probability that the results from an experiment happened by chance. In most cases, a p-value of 0.05 (5%) is accepted to mean the data is valid.

Calculating the Statistic / Test Types

There are three main types of t-test:

  1. An Independent Samples t-test compares the means for two groups.
  2. A Paired sample t-test compares means from the same group at different times (say, one year apart).
  3. A One sample t-test tests the mean of a single group against a known mean.

What is a Paired T Test (Paired Samples T Test / Dependent Samples T Test)?

A paired t test (also called a correlated pairs t-test, a paired samples t test or dependent samples t test) is where you run a t test on dependent samples. Dependent samples are essentially connected — they are tests on the same person or thing. For example:

Knee MRI costs at two different hospitals, Two tests on the same person before and after training, Two blood pressure measurements on the same person using different equipment.

Paired Samples T Test fomula

[t=\frac{\left(\sum D\right) / N}{\sqrt{\frac{\sum D^{2}-\left(\frac{\left(\sum D\right)^{2} )}{N}\right)}{(N-1)(N)}}}]

\(\Sigma D\): Sum of the differences

\(\Sigma D^{2}\): Sum of the squared differences

\((\Sigma D)^{2}\): Sum of the differences, squared.

Find the p-value in the t-table, If you don’t have a specified alpha level, use 0.05 (5%). For example, with df=10, the t-value is 2.228.

Compare your t-table value from above step (2.228) to your calculated t-value (let’s say -2.74). The calculated t-value is greater than the table value at an alpha level of .05. The p-value is less than the alpha level: p <.05. We can reject the null hypothesis that there is no difference between means.

YONG HUANG

YONG HUANG

Hi, I'm Yong Huang. I've recently graduated from Cornell Tech and obtained my master's degree, I shall start my Ph.D. in Computer Science this fall at UC Irvine. Thank you for visiting my site.