To use paired data to construct a confidence interval, the following conditions must be met.
All possible samples of a given size have an equal probability of being chosen; that is, simple random samples are used. The samples are dependent. Both population standard deviations, σ1
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.
In this lesson, you may assume that these conditions are met for all examples and exercises involving paired data.
The value that we want to estimate is the mean of the paired differences for the two populations of dependent data, μd
. Recall that the first step in constructing a confidence interval is to find the point estimate, and the best point estimate for a population mean is a sample mean. Therefore, the mean of the paired differences for the sample data, d⎯⎯
is the point estimate used here.
Formula: Mean of Paired Differences
When two dependent samples consist of paired data, the mean of the paired differences for the sample data is given by d⎯⎯=∑din
is the paired difference for the ith pair of data values and
n is the number of paired differences in the sample data.