The file IUSalariestxt contains the salaries of a random sam

The file IUSalaries.txt contains the salaries of a random sample of 50 IU Bloomington faculty,
along with the salaries of a random 50 IU athletics employees.

\"Job\" \"Salary\"
\"Faculty\" 102416
\"Faculty\" 56333
\"Faculty\" 52934
\"Faculty\" 152893
\"Faculty\" 51090
\"Faculty\" 87998
\"Faculty\" 106930
\"Faculty\" 72500
\"Faculty\" 111628
\"Faculty\" 90945
\"Faculty\" 63500
\"Faculty\" 96593
\"Faculty\" 113825
\"Faculty\" 66086
\"Faculty\" 12565
\"Faculty\" 29088
\"Faculty\" 79698
\"Faculty\" 72828
\"Faculty\" 54631
\"Faculty\" 184296
\"Faculty\" 83209
\"Faculty\" 80937
\"Faculty\" 80123
\"Faculty\" 154000
\"Faculty\" 111180
\"Faculty\" 41820
\"Faculty\" 82430
\"Faculty\" 70000
\"Faculty\" 67170
\"Faculty\" 101536
\"Faculty\" 81723
\"Faculty\" 307668
\"Faculty\" 72930
\"Faculty\" 90000
\"Faculty\" 51425
\"Faculty\" 50245
\"Faculty\" 53987
\"Faculty\" 105305
\"Faculty\" 31761
\"Faculty\" 140273
\"Faculty\" 66900
\"Faculty\" 45900
\"Faculty\" 99000
\"Faculty\" 121190
\"Faculty\" 40216
\"Faculty\" 69750
\"Faculty\" 128312
\"Faculty\" 168943
\"Faculty\" 51201
\"Faculty\" 72500
\"Athletics\" 21107
\"Athletics\" 40800
\"Athletics\" 65000
\"Athletics\" 39814
\"Athletics\" 47861
\"Athletics\" 28434
\"Athletics\" 208539
\"Athletics\" 59022
\"Athletics\" 24480
\"Athletics\" 39541
\"Athletics\" 215701
\"Athletics\" 34272
\"Athletics\" 168300
\"Athletics\" 31824
\"Athletics\" 16084
\"Athletics\" 604858
\"Athletics\" 43410
\"Athletics\" 54060
\"Athletics\" 71400
\"Athletics\" 98838
\"Athletics\" 30805
\"Athletics\" 24149
\"Athletics\" 49939
\"Athletics\" 41820
\"Athletics\" 28267
\"Athletics\" 122400
\"Athletics\" 37170
\"Athletics\" 215701
\"Athletics\" 51154
\"Athletics\" 43697
\"Athletics\" 86626
\"Athletics\" 56100
\"Athletics\" 35714
\"Athletics\" 40810
\"Athletics\" 46818
\"Athletics\" 51000
\"Athletics\" 89573
\"Athletics\" 178500
\"Athletics\" 15267
\"Athletics\" 105000
\"Athletics\" 40522
\"Athletics\" 100717
\"Athletics\" 53040
\"Athletics\" 35714
\"Athletics\" 87277
\"Athletics\" 107100
\"Athletics\" 36683
\"Athletics\" 109936
\"Athletics\" 25210
\"Athletics\" 52530

Do IU Blo omington faculty and IU Athletics employees have the same distribution of salaries?
Explain

Solution

let X denote the salary of IU Blo ominhgton faculty and Y denote the salary of IU Atheletics employees

assuming X follows an absolutely continuous CDF F(x)

and Y follows an absolutely continuous CDF G(x)

here we are to test H0: F(x)=G(x) for all x    vs H1: F(x) not equal to G(x) for at least one x

that is we are to test whether the slaries have the same distribution or not

we have a sample of size 50 from X and a sample of size 50 from Y

here run test is performed

at first all the 100 values of X and Y are mixed up

the median for all these 100 values comes out to be 66493.

let the combined sample be defined as Z

define a function ki=1   if   Zi<66493

                           =0   if Zi>66493       i=1,2,3...............,100

so ki\'s is a sequence of 0\'s and 1\'s.

run is defined as a sequence of 0\'s bounded by 1\'s or vice versa.

[example: 0011110001010. here there are 4 runs of 0\'s and 3 runs of 1\'s. total number of runs=7]

now under H0 X and Y have the same distribution. hence the values should be well mixed up. hence the total number of runs should be large under H0 than under H1

now from this data the number of runs is 50

and p-value is 0.841

performing a 5% level of significance test we vale p-value>0.05

hence we accpet H0 and conclude that both the salaries have the same distribution.