83. Consider numerical observations . It is frequently of interest to know whether the xi s are (at least approximately)

symmetrically distributed about some value. If n is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram.

However, if n is not very large, such pictures are not particularly informative. Consider the following alternative. Let y1 denote the smallest xi, y2 the second smallest xi, and so on.

Then plot the following pairs as points on a two-dimensional coordinate system:

There are n/2 points when n is even and when n is odd.

a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it

(a long upper tail)?

b. The accompanying data on rainfall (acre-feet) from 26 seeded clouds is taken from the article “A Bayesian Analysis of a Multiplicative Treatment Effect in Weather Modification” (Technometrics, 1975: 161–166).

Construct the plot and comment on the extent of symmetry or nature of departure from symmetry.

4.1 7.7 17.5 31.4 32.7 40.6 92.4 115.3 118.3 119.0 129.6 198.6 200.7 242.5 255.0 274.7 274.7 302.8 334.1 430.0 489.1 703.4 978.0 1656.0 1697.8 2745.6

(n 2 1)/2

(yn22 2 x|

, x|

2 y3), c

(yn21 2 x|

, x|

2 y2(y ), n 2 x|

, x| 2 y1), x1,

c, xn x1 5 x1 xt xt2k xt xt, xt21,

c, x1 xx t23 t22 xx t22 t xt21 5 axt21 1 (1 2 a)xt22