Difference Between Two Population Means

Here, we describe estimation and hypothesis-testing procedures for the difference between two population means when the samples are dependent. In a case of two dependent samples, two data values—one for each sample—are collected from the same source (or element) and, hence, these are also called paired or matched samples. For example, we may want to make inferences about the mean weight loss for members of a health club after they have gone through an exercise program for a certain period of time.
To do so, suppose we select a sample of 15 members of this health club and record their weights before and after the program. In this example, both sets of data are collected from the same 15 persons, once before and once after the program. Thus, although there are two samples, they contain the same 15 persons. This is an example of paired (or dependent or matched) samples. The procedures to make confidence intervals and test hypotheses in the case of paired samples are different from the ones for independent samples.
Two samples are said to be paired or matched samples when for each data value collected from one sample there is a corresponding data value collected from the second sample, and both these data values are collected from the same source. As another example of paired samples, suppose an agronomist wants to measure the effect of a new brand of fertilizer on the yield of potatoes. To do so, he selects 10 pieces of land and divides each piece into two portions. Then he randomly assigns one of the two portions from each piece of land to grow potatoes without using fertilizer (or using some other brand of fertilizer).

The second portion from each piece of land is used to grow potatoes with the new brand of fertilizer. Thus, he will have 10 pairs of data values. Then, using the procedure to be discussed in this article, he will make inferences about the difference in the mean yields of potatoes with and without the new fertilizer. The question arises, why does the agronomist not choose 10 pieces of land on which to grow potatoes without using the new brand of fertilizer and another 10 pieces of land to grow potatoes by using the new brand of fertilizer?
If he does so, the effect of the fertilizer might be confused with the effects due to soil differences at different locations. Thus, he will not be able to isolate the effect of the new brand of fertilizer on the yield of potatoes. Consequently, the results will not be reliable. By choosing 10 pieces of land and then dividing each of them into two portions, the researcher decreases the possibility that the difference in the productivities of different pieces of land affects the results.

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New York University
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