How To: Statistics – Two Sample and Paired t-Tests

One thing that regularly stumps scientists is the handling of data. We seem to be very good at generating obscene amounts of it, but representing it meaningfully can be a little off putting if you don’t happen to be a bioinformatician. Let’s continue our tutorial series by introducing Two-Sample t-Tests and Paired t-Tests to see how we can easily incorporate statistical analysis into our work.

Of course, the calculations involved can be done on a simple calculator but your task will be made much easier with the use of spreadsheet software (Excel) or more specialized tools which are available as most high schools and universities (Minitab, Prism, SigmaPlot, etc.). Each will have their own tutorials on carrying out these tests and so this article will not be heavily technical but rather focus on the correct application of statistical testing.

Test When To Use An Example
1 sample t-test Tests if the mean of a single population is equal to hypothesized value A lecturer claims that the mean time taken to complete a quiz is 1 hour. From a sample data set, can we reject this claim?
2 sample t-test Tests if the difference between means of two independent populations is equal to hypothesized value Does the mean quiz score of female students differ significantly from the mean quiz score of male students?
paired t-test Tests if the difference between means of dependent or paired observations is equal to a hypothesized value The mean response time of adults before and after they have consumed alcohol. Is the difference significant enough to conclude that alcohol affects response time?

Main Points to Remember

The t-test is a form of hypothesis testing – to use a set of sample data to test a hypothesis for the entire population.  It is used when the population standard deviation (σ) is unknown and the sample size is small (n<30). In real-world samples we don’t usually have a basis for knowing σ.

Since the t-distribution becomes equivalent to the normal distribution (bell curve) when the sample size is large, the correct practice is:

  • If σ known, use normal distribution.
  • If σ unknown:
    • If n>30, use normal distribution.
    • If n<30, use t-distribution.

Last time out we looked at the one-sample t-test and how we could use it for hypothesis testing in the example ‘determining whether the mean time for completing an online quiz is statistically different from the lecturer’s claim of 1 hour’. If you are unsure of the terminology (μ, x̅, s, σ, etc), confidence intervals or simply get lost, it might be a good idea to check out our How To: One-Sample t-Test first.

Two-Sample t-Test

The two-sample t-test is also used for hypothesis testing, to determine if the means of two independent populations is equal to a target value. For example, “does the mean quiz score of female students (μFS) differ significantly from the mean quiz score of male students(μMS)?” In this case, the hypotheses can be quantified:

  • The null hypothesis (H0): There is no difference in the means (μFS – μMS = 0)
  • The alternative hypothesis (H1): There is a significant difference (μFS – μMS ≠0)

Using software generates a similar set of parameters as a one-sample t-test, but with, well, two samples! The key value here is the difference between sample means, which can then be used to generate a 95% confidence interval, based on this estimate and the standard deviation of the samples. If the 95% confidence interval does not include H(the value 0, in this case) you can reject the null hypothesis and conclude that a difference exists between the two samples.

Paired t-Test

A paired t-test is in fact performing a one-sample t-test on the differences between paired observations. Paired observations are related in some way, such as an individual before and after a certain treatment, or an individual who is subject to similar (hence paired) treatments concurrently.

Formulating the null and hypotheses using the example given above ‘The mean response time of adults before and after they have consumed alcohol’:

  • The null hypothesis (H0): There is no difference before and after (μFS – μMS = 0)
  • The alternative hypothesis (H1): There is a significant difference (μFS – μMS ≠0)

The data generated from a group of 100 people taking a reaction time test 30 minutes after consumption of one ‘standard’ drink (14 g alcohol) would look something like this:

Paired t-test for After – Before

Variable n Mean St Dev 95% CI T-statistic p-value
After  100  1372.6  896.2  –  –  –
Before  100  1272.8  873.7  –  –  –
Difference  100  99.8  245.5 51.1,
 4.06  0.000

From the p-value of 0.000 we can reject the null hypothesis at the 0.05 α-level and conclude that reaction time increased after individuals consumed one ‘standard’ unit of alcohol.

I hope that this tutorial has been useful, happy t-testing!