Now we want to consider the situation where the outcome is continuous, but we are comparing two matched samples. This would be used for example if we had before and after data on each participant. We also saw this situation when we looked at confidence intervals, and once again we focus on the difference in the two measures. We could have serial measures in time at time point 1 and then at time point two, or this might arise on a cross-over trial in which participants are measured on each arm of a clinical trial, say a treatment and a placebo.
The null hypothesis is that the difference is 0. The alternative can be that that the difference is greater than 0, less than 0, or just not equal to 0.
There is a test statistic for the large sample case and t-statistic for the small sample case.
Example: Is there a difference in mean systolic blood pressure measured at exams 6 and 7 (4 years apart) in the Framingham Study? So, we're asking if there was a difference in systolic blood pressures over a 4 year span of time. We're just going to look at 15 participants here. Their mean difference was negative 5.3, and the standard deviation was 12.8. We computed the differences by subtracting the exam 6 measure from the exam 7 measure.
The null hypothesis is that there is no difference, and the alternative hypothesis is that there is a difference, without specifying a direction. We use alpha equal to 0.05, and use a t-statistic because it is a small sample, and degrees of freedom will be n-1 or 14.
The decision rule comes from the table of t-statistics using the two-sided value for alpha=0.05 and 14 degrees of freedom.
So, the critical value is 2.145.
Then compute the test statistic. The sample mean difference is negative 5.3. Mu sub d is the value specified in the null hypothesis, and here it is 0. The denominator is the standard deviation (12.8) divided by the square root of n, which was 15. We get negative 1.60. This falls between our critical values, so we do not reject H0. We do not have statistically significant evidence of a change in mean systolic blood pressure. Note that we're not saying that there was no change in blood pressure. We can only say that we don't have strong evidence of a change.