Paired-samples t-test

Objective

To compare the population mean of two scale-level variables X1 and X2 for the same population

Note:

  1. The test variables X1 and X2 should be scale level and normally distributed.

  2. A paired-samples t-test between X1 and X2 is mathematically equivalent to a one-sample t-test of X2-X1 (or X1-X2) using 0 as test value.

Hypotheses

Two-tailed

  • H0: X1 and X2 have the same population mean.

  • H1: X1 and X2 have different population mean.


One-tailed

  • H0: X1 and X2 have the same population mean.

  • H1: X2 has a larger population mean than X1.

Note : Strictly speaking, H0 for the one-tailed tests should be written as "The population mean of X1 is not greater than / smaller than that of X2", but since we don't care about the other side here, we would use the same H0 as the two-tailed test for simplicity.

Note: It doesn't matter which variable is X1 and which is X2 as far as it is defined clearly in your writing.

Note : We conduct the above one-tailed test only if the sample mean of X2 is already greater than the sample mean of X1.

Example

You want to know if there is any difference between the pre-test (X1) and post-test (X2) results in the population.

Note: The paired variables represent two different variables for the same individuals, e.g. pre-test and post-test results.

Two-tailed

  • H0: The two variables have the same population mean.

  • H1: The two variables have different population mean.

One-tailed

  • H0: X1 and X2 have the same population mean.

  • H1: X2 has a larger population mean than X1.

Procedures

SPSS

  • Analyze -> Compare Means -> Paired-Samples T Test -> Specify the paired variables

Jamovi

  • Analyses -> T-Tests -> Paired Samples T Test -> Specify the paired variables

  • Under Tests, select Student’s for t-test, or Wilcoxon rank for non-parametric test.

  • Under Hypothesis, select the direction of the hypotheses.

  • Optionally, under Additional Statistics, select the descriptive statistics or confidence intervals to be shown. Also do the assumption checks if necessary.

Interpretation

Unless otherwise specified, the p-value given in the software is the two-tailed p-value. Divide it by two to get the one-tailed p-value in case of a one-tailed test.

Two-tailed test:

If p is not <0.05 (or other significant levels), we take H0 as true, i.e., X1 and X2 have the same population mean.

If p<0.05 (or other significant levels), we take H1 as true, i.e., X1 and X2 have different population means.

One-tailed test:

If p is not <0.05 (or other significant levels), we take H0 as true, i.e., X1 and X2 have the same population mean.

If p<0.05 (or other significant levels), we take H1 as true, i.e., X2 has a larger population mean than X1.

What to report

  1. Name of the test being used (paired-samples t-test)

  2. The test variables

  3. The t statistic

  4. The p-value

  5. Your conclusion of the test

  6. Elaboration of the result in your research context