1 Econ 310 Professor Wallace Fall 2013 Lab Section Handout Answers (Week 8) Instructions: Complete the following problem and hand in your solution by the end of lab. Fitting shoes based on arch type and injury prevention. A common procedure for fitting running shoes practiced by running specialty shoe stores and by others has been to recommend motion control shoes to individuals with low arches, stability shoes to individuals with normal arches, cushioned shoes to individuals with high arches. In a recent study, Knapik et al. (2010) investigate whether this approach to assigning shoes influences the incidence of running related injuries among US Air Force recruits.1 In their study US Air Force basic training recruits were randomly assigned to an experimental group or a control group. The experimental group had shoes assigned on the basis of arch type; all members of the control group 1 were assigned stability shoes, regardless of arch type, and all members of control group 2 were allowed freely choose which shoe to wear on the basis of fit, comfort, aesthetics, or any other criteria. Knapik et al. report the following results with respect to the Training Injury Index (TII), an index of lower extremity injuries, and the Overuse Injury Index (OII), an index of overuse injuries Experimental Group (n=800) Control Group 1 (n=700) Control Group 2 (n=750) Sample mean Sample std. Sample mean Sample std. Sample mean Sample std. Training Injury Index 4.65 10.71 3.92 10.66 3.75 10.65 Overuse Injury Index 5.86 10.85 5.25 10.70 5.15 10.62 A) Provide an estimate of the impact of being assigned shoes on the basis of arch type versus freely choosing shoes (control group 2) on the TII. There are effectively three hypothetical populations in this experiment. The first is the population that would exist when all Air Force recruits assigned to shoes on the basis of their arch type. The mean of this population is experimental e. The other populations are those that would exist when Air Force recruits are assigned stability shoes (mean 1cg) and when Air Force recruits are allowed to choose shoes on their own (mean 2)cg. This question is asking for a point estimate of 2e cg    , or the difference in means of the TII between the world where all recruits are assigned shoes on the basis of arch types and the world where they pick their own shoes. An unbiased estimator for this parameter is 24.65 3.75 0.90e cgx x x     . 1 American Journal of Preventive Medicine (2010), Vol 38(1S), pp. S197-S211.2 B) Construct a 95% confidence interval for the effect of being assigned shoes on the basis of arch type versus freely choosing shoes (control group 2) on the TII.   22222221 100% : 10.71 10.6595% : 0.90 1.96800 75095% : 0.90 1.06 1.16,1.96cgee cgssCI x ZnnCICI         C) With no priors as to the effect of shoe assignment on injury risk, what is the most appropriate hypothesis for Knapik et al. to evaluate. Use control group 2 as the control group. 022:0:0e cgA e cgHH D) Evaluate the hypothesis test specified in part C) using the data provided on the TII and significance level of 0.05. What can you conclude?3 E) Suppose that being assigned shoes on the basis of arch type reduces the TII by 0.5. i. Provide a definition of statistical power in the context of this problem. Statistical power is the probability of correctly rejecting the null hypothesis 02:0e cgH given that 0.5   (i.e., the probability of correctly rejecting the null given that assigning recruits shoes based on arch type reduces the TII by 0.5) ii. Find the values of the sample statistic lead to rejection of the null. Be sure to clearly state any assumptions that you make.4 iii. Draw a picture in which you show the distribution under the null and the true (alternate) distribution. Label the regions of power and type 2 error in the true (alternate) distribution. Provide an eyeball guess as to the level of statistical power associated with your test in part C.5 iv. Compute the statistical power of the test in part C)6 v. How would the statistical power of the test in part C. change if you changed the significance level to 0.01? Increased the sample sizes? Assumed that assigning shoes based on arch type decreased the TII by 1? Assumed that assigning shoes on the basis of arch type injury increases the TII by 0.5? For this question you do not need to provide precise values – just directions of change. Be sure to explain your answers. If we increased the significance level from 0.01 power would increase. By increasing the significance level we effectively move in the values of cirtx and critx so they closer to the mean of the distribution under the null hypothesis. This means that that there has to be a greater probability of drawing values more extreme that cirtx and critx from the true distribution. If we increase the sample sizes the distribution under the null and the true distribution will become more compressed about their means. This will have the effect of moving the values of cirtx and critx so they are closer to the mean of the distribution under the null. It will also concentrate values in the true distribution so that are greater proportion are closer to the mean. Both of these changes will lead to increased power. If we assume that assigning shoes on the basis of arch type reduces the TII by 1 (instead of 0.5) the power of the test conducted in part C) would increase because power almost always increases with the difference between means of the null and the true (alternate) distributions all else equal. The exception to this statement occurs when the power is already maxed out at 1. In this case increasing the difference in the mean X between the null and true (alternate) distribution would not result in any increase in power. If assigning shoes on the basis of arch type increases the TII by 0.5, the power will remain the same. Under this scenario the difference between the means in the distribution under the null and the true (alternate) distributions is the same as it is in the original problem and all other parameters are the same. Therefore, the power has to be the same for this two tailed test. The only difference is that in the original question most of the power is in the left