In other words if group 1 is 2 points higher than

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In other words, if group 1 is 2 points higher than group 2 on the log scale ( P 1 = P 2 + 2), then group 1 is 10 P 2+2 = 10 P 2 (100) or 100 times higher in the original scale. Therefore, once we apply the back-transformation, we really have an interval of plausible values for the multiplicative change in the median of the response variable. (n) Perform the back-transformations and interpret the new confidence interval in terms of a ratio of treatment medians. Study Conclusions The rainfall distributions reveal that the rainfall amounts from the seeded clouds have a larger mean and larger variability than from unseeded clouds, and both groups have very right-skewed distributions of rainfall with several outliers. A two-sample t -test may not be valid for these data so we could consider a randomization test on the medians, but then we don’t have a convenient way to find a confidence interval. This is a situation where a ln-transformation is likely to be helpful in rescaling the data so that the t -procedures are more valid. Examining graphs of the transformed data confirm that the variability is now more similar and both shapes are symmetric. The two-sample t -test applied to the ln-transformed data can thus be used as an approximation to the randomization test. Also, the confidence interval for the difference in the ln rainfall can be back-transformed to provide a statement about the multiplicative treatment effect of cloud seeding: We are 95% confident that the median volume of rainfall on days when clouds are seeded is 1.3 to 7.7 times larger when clouds are seeded as opposed to when they are not seeded. Because random assignment was used to assign the clouds to the seeding conditions, it is safe to interpret this as evidence that the seeding causes larger rainfall amounts on average. Although the transformation again gives us a comparison of medians, which is often preferable with skewed data and/or outliers, keep in mind, sometimes we really prefer the parameter to be a mean than a median, such as when you want to focus on the “total” amount (population mean × population size). Practice Problem 4.7 Compare these two analyses to a two-sample t -test and a two-sample t -confidence interval on the original data.
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Chance/Rossman, 2015 ISCAM III Investigation 4.8 288 SECTION 4: MATCHED PAIRS DESIGNS In this section, you will again compare two sets of quantitative observations but with one key difference in how the data were collected. You will see advantages to collecting data with a “paired” design, and then you will investigate how to make the corresponding changes in the appropriate analyses. Investigation 4.8: Chip Melting Times Suppose you want to compare the melting times of semisweet chocolate chips to peanut butter chips using the students in your class. To melt the chips, one chip will be placed on each student’s tongue and then held against the roof of his or her mouth. The time until the chip is completely melted, without any “encouragement” by the student, will be recorded (in seconds).
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