R provide an interpretation of this p value in

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(r) Provide an interpretation of this p-value in context (make sure you address the statistic, the source of the randomness in the study, and what you mean by “more extreme”). Conclusions (s) Significance: What conclusion would you draw from this simulation analysis regarding the question of whether the learning improvements in the sleep deprived group are significantly lower (on average) than those in the unrestricted sleep group? Also explain the reasoning process by which your conclusion follows from the simulation results. (t) Causation: Does the design of this study allow you to conclude that the reduction in improvement scores is caused by the sleep deprivation? Explain. (u) Generalizability: To what “population” is it reasonable to generalize these results? Exact p-value? Although not as convenient as before, we can determine the exact p-value for this randomization test by considering all the different possible random assignments of these 21 subjects into groups of 11 and 10, determining the difference in means (or medians) for each, and then counting how many are at least as extreme as the observed difference. The following histogram (produced using the combn function in R, from the combinat package, assuming imprvs contains the response variable values) shows the distribution of all 352,716 possible differences in group means.
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Chance/Rossman, 2015 ISCAM III Investigation 4.4 276 indices = 1:21 allcombs = combn(21, 10) diffs = 1:ncol(allcombs) for (i in 1:ncol(allcombs)){ group1=imprvs[allcombs[,i]] group2=imprvs[setdiff(indices, allcombs[,i])] diffs[i]=mean(group1)-mean(group2) } hist(diffs) abline(v=15.92, col=2) approx. run time: 1 min (v) Do your simulation results reasonably approximate this null distribution? (w) It turns out that 2456 of the 352,716 different random assignments produce a difference in group means of 15.92 milliseconds or larger. Use this information to determine the exact p-value of this randomization test. Is the approximate p-value from your simulation close? Discussion: The exact randomization distribution consists of every possible random assignment and calculates the statistic of interest (e.g., difference in means) for each one. Then we simply count how many of the configurations result in a value of the statistic at least as extreme (as defined by the alternative hypothesis) as the actual observed result. As you might expect, it can be extremely tedious, even with computers, to list out all of these possible random assignments. And these group sizes are relatively small! One shortcut is to only count how many assignments give results more extreme than the one observed, but as you will see we can often appeal to a mathematical model as well. Note: You can carry out these simulations in R or Minitab as well, see the Technology Detour on the next page.
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Chance/Rossman, 2015 ISCAM III Investigation 4.4 277 Technology Detour Simulating a Randomization Test for a Quantitative Response In Comparing Groups (Quantitative) applet Copy and paste data (match the ordering of the explanatory and response variables) and press Use Data.
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