1-1 Statistical and Critical Thinking 5 TABLE 1-1 Shoe Print Lengths and Heights of Men Shoe Print (cm) 27.6 29.7 29.7 31.0 31.3 31.4 31.8 34.5 Height (cm) 172.7 175.3 177.8 175.3 180.3 182.3 177.8 193.7 Source of the Data The second step in our preparation is to consider the source (as indicated in Figure 1-3). The data in Table 1-1 are from Data Set 9 “Foot and Height” in Appendix B, where the source is identified. The source certainly appears to be reputable. Conclude 1. Significance • Do the results have statistical significance? • Do the results have practical significance? Analyze 1. Graph the Data 2. Explore the Data • Are there any outliers (numbers very far away from almost all of the other data)? • What important statistics summarize the data (such as the mean and standard deviation described in Chapter 3)? • How are the data distributed? • Are there missing data? • Did many selected subjects refuse to respond? 3. Apply Statistical Methods • Use technology to obtain results. Prepare 1. Context • What do the data represent? • What is the goal of study? 2. Source of the Data • Are the data from a source with a special interest so that there is pressure to obtain results that are favorable to the source? 3. Sampling Method • Were the data collected in a way that is unbiased, or were the data collected in a way that is biased (such as a procedure in which respondents volunteer to participate)? FIGURE 1-3 Statistical and Critical Thinking Survivorship Bias In World War II, statistician Abraham Wald saved many lives with his work on the Applied Mathematics Panel. Military leaders asked the panel how they could improve the chances of aircraft bombers returning after missions. They wanted to add some armor for protection, and they recorded locations on the bombers where damaging holes were found. They reasoned that armor should be placed in locations with the most holes, but Wald said that strategy would be a big mistake. He said that armor should be placed where returning bombers were not damaged. His reasoning was this: The bombers that made it back with damage were survivors, so the damage they suffered could be survived. Locations on the aircraft that were not damaged were the most vulnerable, and aircraft suffering damage in those vulnerable areas were the ones that did not make it back. The military leaders would have made a big mistake with survivorship bias by studying the planes that survived instead of thinking about the planes that did not survive. Sampling Method Figure 1-3 suggests that we conclude our preparation by considering the sampling method. For the data in Table 1-1, individuals were randomly selected, so the sampling method appears to be sound. Sampling methods and the use of random selection will be discussed in Section 1-3, but for now, we stress that a sound sampling method is absolutely essential for good results in a statistical study. It is generally a bad practice to use voluntary response (or self-selected) samples, even though their use is common. and heights of males. This goal suggests a reasonable hypothesis: Males with larger shoe print lengths tend to be taller. (We are using data for males only because 84% of burglaries are committed by males.)
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