Two Dependent Samples (Matched Pairs)

Introduction to Dependent Samples

In statistics, dependent samples (also called matched pairs) refer to data sets where each observation in one sample can be paired with a related observation in another sample. This is in contrast to independent samples, where observations are unrelated. Matched pairs are commonly used in before-and-after studies, twin studies, or any scenario where measurements are naturally paired.

• Dependent samples arise when the same subjects are measured twice, or when subjects are matched in pairs based on certain criteria.

• Analysis of matched pairs focuses on the differences between paired observations.

• Common applications include medical trials, educational interventions, and comparative studies.

Designing Experiments with Matched Pairs

When designing experiments or observational studies, using matched pairs can control for confounding variables and increase statistical power. The key is to ensure that the pairing is meaningful and relevant to the research question.

• Example 1: SAT scores are measured for students before and after a training program. Each student's before and after scores form a matched pair.

• Example 2: Ages of actors and actresses listed by year are paired to compare age differences.

Objectives of Matched Pairs Analysis

• Hypothesis Test: Test claims about the mean of the population of all differences.

• Confidence Interval: Estimate the mean of the population of all differences.

Requirements for Matched Pairs Analysis

• The sample is a simple random sample.

• The two samples are dependent (matched pairs).

• If , or the differences are normally distributed, the test is robust.

Additional info: For small samples (), normality of the differences should be checked.

Notation

• = difference between two values of a matched pair

• = mean value of the differences , for the population of all matched pairs

• = mean value of the differences , for the paired sample data

• = sample standard deviation of the differences

• = number of pairs of sample data

Hypothesis Testing for Matched Pairs

Test Statistic

The test statistic for matched pairs is:

• P-value: Calculated using statistical software or calculator.

• Critical Value: Found using t-distribution tables or calculator.

Confidence Interval Estimate

The confidence interval for the mean difference is:

Where the margin of error is:

Hypothesis Test Procedure

• Null hypothesis:

• Alternative hypothesis: (two-tailed), , or (one-tailed)

• P-values, critical values, and confidence intervals all provide the same conclusion for matched pairs.

Steps for Inference with Matched Pairs

1. Verify data comes from matched pairs and requirements are satisfied.

2. Find the difference () for each pair of sample values (always subtract in the same way).

3. Find the mean and standard deviation of the differences.

4. Use the procedures for hypothesis testing for single population means (Test or Interval) for the data found in step 3.

Calculator Instructions (TI-83/84 Plus)

• Do not use the menu item 2-SampTTest for matched pairs; use T-Test.

• Enter the data for the first variable in L1 and the second variable in L2.

• Create a list of differences in L3 by entering L1 - L2.

• Proceed with T-Test using the list of differences.

Example: Comparing Ages of Actors and Actresses

Data Table

Additional info: The differences are calculated as Actress - Actor for each pair.

Hypothesis Test for the Example

• Null hypothesis:

• Alternative hypothesis: (testing if actresses are younger than actors)

• Calculate differences, mean , and standard deviation .

• Use T-Test to find p-value and confidence interval.

• If p-value < 0.05, reject and conclude actresses are younger.

Result: Since all differences are negative and p-value is 0.014 < 0.05, we reject and conclude that best actresses are younger than best actors.

Confidence interval: (-16.5, -2.9) does not include 0, supporting the conclusion.

Summary Table: Matched Pairs vs. Independent Samples