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# The Matched-Pairs Design for Comparing Two Treatment Means Study Guide (page 4)

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Updated on Oct 5, 2011

#### Step 2: Verify Necessary Conditions for a Test and, if Satisfied, Construct the Test Statistic

Although the runners were recruited, the order in which the treatments (newly developed shoes and favorite shoes) were observed was randomly determined. Thus, the observed differences are a random sample of all possible differences for these 24 runners, and the first condition for inference is satisfied. Often, race times are normally distributed, so the differences in race times under two treatments would be normally distributed. The sample size is not sufficient to test the assumption of normality rigorously. However, from inspection of the graphs and summary statistics earlier in this lesson, it is not unreasonable to assume that the differences in run times using the favorite and new shoes are normally distributed. Thus, the conditions for inference are assumed to be satisfied.

Because the study has a paired design, the test statistic is the now familiar form

.

#### Step 3: Find the p-Value Associated with the Test Statistic

If the null hypothesis is true, the test statistic has a t-distribution with (n – 1) = 23 degrees of freedom. Given the alternative hypothesis, we want to reject the null hypothesis if tT gets too large. Thus, p = P(t > 3.27). In the t-table on the line for 23 degrees of freedom, 3.27 lies between 2.807 and 3.485, corresponding to upper tail probabilities of 0.005 and 0.001, respectively; thus, 0.001 < p < 0.005.

#### Step 4: Decide Whether or Not to Reject the Null Hypothesis

The p-value observed in this study indicates that a test statistic of this magnitude is very unusual if the null hypothesis is true. Therefore, we reject the null hypothesis and decide in favor of the alternative.

#### Step 5: State Conclusions in the Context of the Study

The mean time for a 100-meter race was significantly less when runners wore the newly developed shoes compared to their favorite running shoes.

## The Matched-Pairs Design for Comparing Two Treatment Means In Short

Designs comparing two treatments or populations have been discussed. The matched-pairs design allows one to account for known or suspected sources of variability in the design. The two-group design is useful when a reasonable basis for pairing is not available or feasible. For the paired design, confidence intervals and hypothesis tests on the difference in the treatment means were described.

Find practice problems and solutions for these concepts at The Matched-Pairs Design for Comparing Two Treatment Means Practice Exercises.

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