When testing the difference of means for paired data using a paired t-test, the null hypothesis is that the average of the differences between the paired observations is zero. In other words, there is no significant difference in the means of the two related groups
. Formally, if di=x2i−x1id_i=x_{2i}-x_{1i}di=x2i−x1i represents the difference between the paired observations for subject iii, the null hypothesis is:
H0:μd=0H_0:\mu_d =0H0:μd=0
where μd\mu_d μd is the population mean of the differences. The alternative hypothesis can be two-tailed or one-tailed, depending on the research question:
- Two-tailed: Ha:μd≠0H_a:\mu_d \neq 0Ha:μd=0 (mean difference is not zero)
- One-tailed: Ha:μd>0H_a:\mu_d >0Ha:μd>0 or Ha:μd<0H_a:\mu_d <0Ha:μd<0 (mean difference is greater or less than zero)
The paired t-test evaluates whether the observed mean difference is statistically significantly different from zero, accounting for the paired nature of the data
Summary:
- Null hypothesis (H0): The mean difference between paired observations is zero.
- Alternative hypothesis (Ha): The mean difference between paired observations is not zero (or is greater/less than zero for one-tailed tests).
This hypothesis reflects that any observed differences are due to random variation if the null is true
