Randomized comparative experiment and not an

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randomized comparative experiment and not an observational study, we can draw a causal conclusion that the sleep deprivation was the cause of the lower learning improvements. However, the subjects were college-aged volunteers, so we may not want to generalize these results to a much different population. Practice Problem 4.4A Explain how the simulation analysis conducted in this section differs from that of the previous section. For example: What is the random process being simulated? What assumptions are underlying the simulation? Practice Problem 4.4B (a) Produce numerical and graphical summaries for the data in FakeSleepDeprivation.txt , representing a new set of 21 responses for the sleep deprivation study. Comment on how the shapes, centers, and variability for the two distributions compare between these data and the original data. (b) Use technology (see Technology Detour on previous page) to carry out a randomization test to compare the improvements for the sleep deprived and unrestricted sleep groups using the hypothetical data. Indicate how you approximated the p-value. (c) How does the p-value for the hypothetical data compare to the p-value for the original data? Explain why this makes sense based on what you learned about how the data sets compared in (a).
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Chance/Rossman, 2015 ISCAM III Investigation 4.5 279 Investigation 4.5: Lingering Effects of Sleep Deprivation (cont.) Reconsider the exact randomization distribution of the differences in sample means for the Sleep Deprivation study. Previously you determined simulation-based p-values and the exact p-value. Now we will explore modeling the randomization distribution of the difference in sample means with a probability distribution. (a) Does this distribution appear to be well modeled by a normal distribution? (b) Suppose you want to use the normal model to approximate the p-value for obtaining a difference in means of 15.92 or larger under the null hypothesis or to compute a confidence interval. What other information do you need to know? To use the normal distribution, we need a measure of the variability in the randomization distribution. In Investigation 4.2, we found that when we have independent random samples from infinite populations we can use 2 2 2 1 2 1 2 1 ) ( n n X X SD V V ² ± where ࠵? represents the “population” standard deviation. Because those V values will almost surely be unknown, we can estimate them using the sample standard deviations, producing the “standard error” of the difference in sample means: 2 2 2 1 2 1 2 1 ) ( n s n s X X SE ² ± (c) For the sleep deprivation data, the sample standard deviations are: s unrestricted = 14.72 ms and s deprived = 12.17 ms Use these values to compute the standard error for the difference in sample means. Compare this to the standard deviation you observed in the applet. In particular, have we over- or under- estimated the variability?
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