Means: paired samples

In the case of two dependent samples, two data values-one in each sample- are collected from the same source and these are called paired or matched pairs.

Suppose that n matched pairs of observations, denoted by

, are selected from two populations with means and .

Our aim is to find confidence interval for .

To find interval estimation we apply following steps:

1. Find n differences

2. Find

3. Calculate

If the population distribution of differences is assumed to be normal, then

confidence interval for the difference between means is given by

where is the number for which

The random variable has a Student’s t distribution with (n -1) degrees of freedom.

Example:

A company claims that its special exercise program significantly reduces weight. A random sample of seven persons were put on exercise program. The following table gives the weights (in kg) of those seven persons before and after the program

Before
After

Make a 95 % confidence interval for the mean of the population paired differences. Assume that the population of paired differences is (approximately) normally distributed.

Solution:

Let d be the difference between the weights before and after the program.

The necessary calculations are shown in the following table

Before	After	Difference
		-2

The values of and are calculated as follows:

.

Then

and

.

In the end, 90 % confidence interval for is

Thus, we can state with 90 % confidence that the mean difference between the weights before and after exercise program is between 2.6 and 8.82 kg.

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