Consider numerical observations . It is frequentlyof interest to know whether the xi s

Chapter 1, Problem 83

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Consider numerical observations . It is frequentlyof interest to know whether the xi s are (at least approximately)symmetrically distributed about some value. If n isat least moderately large, the extent of symmetry can beassessed from a stem-and-leaf display or histogram.However, if n is not very large, such pictures are not particularlyinformative. Consider the following alternative. Let y1denote the smallest xi, y2 the second smallest xi, and so on.Then plot the following pairs as points on a two-dimensionalcoordinate system:There are n/2 points when n is evenand when n is odd.a. What does this plot look like when there is perfect symmetryin the data? What does it look like when observationsstretch out more above the median than below it(a long upper tail)?b. The accompanying data on rainfall (acre-feet) from 26seeded clouds is taken from the article A BayesianAnalysis of a Multiplicative Treatment Effect in WeatherModification (Technometrics, 1975: 161166).Construct the plot and comment on the extent of symmetryor nature of departure from symmetry.4.1 7.7 17.5 31.4 32.7 40.6 92.4115.3 118.3 119.0 129.6 198.6 200.7 242.5255.0 274.7 274.7 302.8 334.1 430.0 489.1703.4 978.0 1656.0 1697.8 2745.6