Example (which tire lasts longer?) To determine whether a new steel-belted radial tire lasts longer than a current model, the manufacturer designs the following experiment. –A pair of newly designed tires are installed on the rear wheels of 20 randomly selected cars. –A pair of currently used tires are installed on the rear wheels of another 20 cars. –Drivers drive in their usual way until the tires wear out. –The number of miles driven by each driver were recorded. See data next. Matched Pairs Experiment

Solution Compare two populations of quantitative data. The parameter is  1 -  2 11 22 The hypotheses are: H 0 : (  1 -  2 ) = 0 H a : (  1 -  2 ) > 0 Mean distance driven before worn out occurs for the new design tires Mean distance driven before worn out occurs for the existing design tires

The hypotheses are H 0 :  1 -  2 = 0 H 1 :  1 -  2 > 0 The test statistic is We run the t test, and obtain the following Excel results. We conclude that there is insufficient evidence to reject H 0 in favor of H 1.

Example continued (using the matched pairs approach) to eliminate variability among observations within each sample the experiment was redone. –One tire of each type was installed on the rear wheel of 20 randomly selected cars (each car was sampled twice, thus creating a pair of observations). –The number of miles until wear-out was recorded Matched Pairs Experiment

Solving by hand –Calculate the difference for each pair. –Calculate the average differences and the standard deviation of the differences. –Build the statistics as follows: –Run the hypothesis test using t distribution with n D - 1 degrees of freedom.

–The hypotheses test for this problem is H 0 :  D = 0 H 1 :  D > 0 The statistic is The rejection region is: t > t  with d.f. = 20-1 = 19. If  =.05, t.05,19 = Since > 1.729, there is sufficient evidence in the data to reject the null hypothesis in favor of the alternative hypothesis. Conclusion: At 5% significance level the new type tires last longer than the current type. See file Tires.xls

Estimating the mean difference

Checking the required conditions for the paired observations case The validity of the results depends on the normality of the differences.

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