# Question: There are many running shoes on the market of varying

There are many running shoes on the market of varying degrees of quality. Long-distance runners require a shoe that provides a significant reduction in impact shock compared to the standard running shoe intended for weekend joggers. A runners’ magazine commissioned a study to evaluate three brands of shoes that claim to provide a reduction in impact shock. Ten experienced long-distance runners were selected to participate in the study. The study would consist of placing sensors in the runners’ shoes to measure impact forces as the runner ran on a treadmill set at a speed of 4 meters per second. Because the impact force is very dependent on the weight and individual stride of the runner, each of the 10 runners will be observed while using all three brands and a widely sold brand that will serve as a control. The runners were evaluated wearing the four brands in a random order with sufficient time between evaluations to allow the runners to be well rested prior to each evaluation. The impact forces (in Newtons) are presented in the following table with the following notation: BC = control brand and B1, B2, and B3 = three new brands.
a. Is there significant (a 5 0.05) evidence of a difference in the four brands of shoes with respect to their mean peak force?
b. How many runners would be needed to conduct this study as a completely randomized experiment? What would be the gains and losses in conducting the study as a completely randomized design?
c. What conditions are necessary in order for the test conducted in part ( a) to pro-vide valid p-values?
d. What is the population to which the results of this study can be validly applied? 18.5 Crossover Designs

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