If there are n pairs of data values and the population distribution of the paired differences is approximately normal, then the sampling distribution for the sample statistic d⎯⎯ follows a t-distribution with n, n−1 degrees of freedom. Hence, the formula for the margin of error is as follows. This is the same formula that is used when estimating a single population mean when σ is unknown. This is because we use the paired differences as a single set of sample data rather than using the data from the two samples separately when working with paired data.
Margin of Error of a Confidence Interval for the Mean of the Paired Differences for Two Populations ( σ Unknown, Dependent Samples)
When both population standard deviations are unknown, the samples taken are dependent, simple random samples of paired data, and either the number of pairs of data values in the sample data is greater than or equal to 30
or the population distribution of the paired differences is approximately normal, the margin of error of a confidence interval for the mean of the paired differences for two populations is given by
is the critical value for the level of confidence, c=1−α such that the area under the t-distribution with n−1 degrees of freedom to the right of tα2/ is equal to α2
is the sample standard deviation of the paired differences for the sample data, an
n is the number of paired differences in the sample data.