Approach

We have seen that the problem of testing whether two paired measurements, X and Y, have equal means is done in terms of the differences

D = Y - X

The test is then expressed as

H0:   µD = 0

HA:   µD ≠ 0

or a one-tailed variant. This is a standard univariate hypothesis test of the form analysed in the previous section.

Paired t-test

The hypotheses are therefore assessed with a standard t-test. The test statistic is

and it is compared against a t distribution with n - 1 degrees of freedom to find the p-value.

Age when children with heart disease start to talk

A research worker studying congenital heart disease wishes to compare the development of cyanotic children with normal children. Among the measurement of interest is the age at which the children speak their first word.

Rather than independently selecting a sample of cyanotic and a sample of normal children for this comparison, the researcher chooses cyanotic children who have a sibling of the same gender. Measurements from the children therefore form a paired data set. The ages in months when the first word was spoken are shown below.

Pair of
siblings
Cyanotic
sibling
Normal
sibling
Difference
(cyanotic - normal)
1
2
3
4
5
6
7
8
9
10
11.8
20.8
14.5
9.5
13.5
22.6
11.1
14.9
16.5
16.5
9.8
16.5
14.5
15.2
11.8
12.2
15.2
15.6
17.2
10.5
2.0
4.3
0.0
-5.7
1.7
10.4
-4.1
-0.7
-0.7
6.0

The data are paired since the measurements come in pairs from the same families.

The researcher wants to test whether cyanotic children speak their first word later on average than children without the disease, so a one-tailed test is appropriate. Denoting the difference within each sibling pair (cyanotic – normal) by D, we are looking for evidence that µD > 0 (meaning that the mean age of speaking the first word is greater for cyanotic children than normal ones). The hypotheses are therefore:

H0:   µD = 0

HA:   µD > 0

Since the p-value for the test is well above zero (0.201), there is no evidence from these data that the cyanotic children learn to speak later.


Select Modified Data from the pop-up menu, then use the slider to investigate how much higher the mean for the 'cyanotic' values would need to be to give strong evidence of a difference.

Blood pressure and the pill

In this example, the blood pressure of 15 college-aged women was measured before starting to take the pill and after 6 months of use. The data were tabulated at the start of this section and are graphed below.

Here we use a two-tailed test as the pill might either increase or decrease blood pressure. Denoting the difference in blood pressure for each woman as D = (after - before), we are therefore interested in the hypotheses:

H0:   µD = 0

HA:   µD ≠ 0

The p-value for this test is calculated on the right below.

The resulting p-value is very small, giving strong evidence that blood pressure has changed. The test only gives evidence of a difference in the mean blood pressures. However the negative t value suggests µD < 0, so it is valid to conclude that there is evidence of a decrease in blood pressure after taking the pill.

Again, select Modified Data and investigate how different the sample means must be to give evidence of a difference in the population means.