Consider numerical observations x1,, xn. It is

Chapter 1, Problem 83

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Consider numerical observations x1,, xn. 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 y1 denote the smallest xi, y2 the second smallest xi, andso on. Then plot the following pairs as points on atwo-dimensional coordinate system: (yn 2 ,x, ,x 2 y1),(yn21 2 ,x, ,x 2 y2), (yn22 2 ,x, ,x 2 y3), There are n/2points when n is even and (n 2 1)y2 when n is odd.a. What does this plot look like when there is perfectsymmetry in the data? What does it look like whenobservations stretch out more above the median thanbelow it (along upper tail)?b. The accompanying data on rainfall (acre-feet) from26 seeded clouds is taken from the article ABayesian Analysis of a Multiplicative TreatmentEffect in Weather Modification (Technometrics,1975: 161166). Construct the plot and comment onthe extent of symmetry or nature of departure fromsymmetry.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