In the table below, X1 is a random sample of 20 observations from an expo- nential population with parameter β = 1.44, so that the median, η = 1. X2 is the same data set plus a constant, 0.6, and random Gaussian noise with mean μ = 0 and standard deviation σ = 0.15.
|
X1
1.26968
|
X2
1.91282
|
X1
1.52232
|
X2
2.17989
|
|
0.28875
|
1.13591
|
1.45313
|
2.11117
|
|
0.07812
|
0.72515
|
0.65984
|
1.45181
|
|
0.45664
|
1.19141
|
1.60555
|
2.45986
|
|
0.68026
|
1.34322
|
0.08525
|
0.43390
|
|
2.64165
|
3.18219
|
0.03254
|
0.76736
|
|
0.21319
|
0.88740
|
0.75033
|
1.16390
|
|
2.11448
|
2.68491
|
1.34203
|
2.01198
|
|
1.43462
|
2.16498
|
1.25397
|
1.80569
|
|
2.29095
|
2.84725
|
3.16319
|
3.77947
|
(i) Consider the "postulate" that the median for both data sets is η0 = 1 (which, of course, is not true). Generate a table of signs, Dmi , of deviations from this postulated median.
(ii) For each data set, determine the test statistic, T +. What percentage of the observations in each data set has plus signs? Informally, what does this indicate about the possibility that the true median in each case is as postulated?