5.6.2 - Inference for Paired Means

When we developed the inference for the independent samples, we depended on the statistical theory to help us. The theory, however, required the samples to be independent. What can we do when the two samples are not independent, i.e., the data is paired?

Consider an example where we are interested in a person’s weight before implementing a diet plan and after. Since the interest is focusing on the difference, it makes sense to “condense” these two measurements into one and consider the difference between the two measurements. For example, if instead of considering the two measures, we take the before diet weight and subtract the after diet weight. The difference makes sense too! It is the weight lost on the diet.

When we take the two measurements to make one measurement (i.e., the difference), we are now back to the one sample case! Now we can apply all we learned for the one sample mean to the difference (Cool!)

Hypothesis Test for the Difference of Paired Means, \(μ_d\) Section

In this section, we will develop the hypothesis test for the mean difference for paired samples. As we learned in the previous section, if we consider the difference rather than the two samples, then we are back in the one-sample mean scenario.

The possible null and alternative hypotheses are:

\(H_0\colon \mu_d=0\)


We still need to check the conditions and at least one of the following need to be satisfied:

  • The differences of the paired follow a normal distribution
  • The sample size is large, \(n>30\).

Test Statistics:

If at least one condition is satisfied then...


Will follow a t-distribution with \(n-1\) degrees of freedom.

The same process for the hypothesis test for one mean can be applied. The test for the mean difference may be referred to as the paired t-test or the test for paired means.

Minitab 18


Paired t-Test Section

You can use a paired t-test in Minitab to perform the test. Alternatively, you can perform a 1-sample t-test on difference = before and after diet plan.

  • Choose Stat > Basic Statistics > Paired t
  • Click Options to specify the confidence level for the interval and the alternative hypothesis you want to test. The default null hypothesis is 0.

Diet Plan Section

The Minitab output for paired T for before-after diet plan is as follows: 


95% lower bound for mean difference: 0.0505 

T-Test of mean difference = 0 (vs > 0): T-Value = 4.86 P-Value = 0.000 

Using the p-value to draw a conclusion about our example:

p-value = \(0.000 < 0.05\)

Reject \(H_0\) and conclude that before diet weight is greater than after diet weight.