29 8.1 Inference for Two Dependent Samples (Matched Pairs)
29 8.1 Inference for Two Dependent Samples (Matched Pairs)
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Learning Objectives
By the end of this chapter, the student should be able to:
- Classify hypothesis tests by type
- Conduct and interpret hypothesis tests for two population means, population standard deviations known
- Conduct and interpret hypothesis tests for two population means, population standard deviations unknown
- Conduct and interpret hypothesis tests for matched or paired samples
- Conduct and interpret hypothesis tests for two population proportions
Studies often compare two groups. For example, maybe researchers are interested in the effect aspirin has in preventing heart attacks. One group is given aspirin and the other a placebo, and the heart attack rate is studied over several years. Other studies may compare various diet and exercise programs. Politicians compare the proportion of individuals from different income brackets who might vote for them. Students are interested in whether SAT or GRE preparatory courses really help raise their scores.
You have learned to conduct inference on single means and single proportions. We know that the first step is deciding what type of data we are working with. For quantitative data we are focused on means, while for categorical we are focused on proportions. In this chapter we will compare two means or two proportions to each other. The general procedure is still the same, just expanded. With two sample analysis it is good to know what the formulas look like and where they come from, however you will probably lean heavily on technology in preforming the calculations.
To compare two means we are obviously working with two groups, but first we need to think about the relationship between them. The groups are classified either as independent or dependent. Independent samples consist of two samples that have no relationship, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Dependent samples consist of two groups that have some sort of identifiable relationship.
Two Dependent Samples (Matched Pairs)
Two samples that are dependent typically come from a matched pairs experimental design. The parameter tested using matched pairs is the population mean difference. When using inference techniques for matched or paired samples, the following characteristics should be present:
- Simple random sampling is used.
- Sample sizes are often small.
- Two measurements (samples) are drawn from the same pair of (or two extremely similar) individuals or objects.
- Differences are calculated from the matched or paired samples.
- The differences form the sample that is used for analysis.
To perform statistical inference techniques we first need to know about the sampling distribution of our parameter of interest. Remember although we start with two samples, the differences are the data we are interested in and our parameter of interest is μd, the mean difference.