# Kruskal-Wallis Nonparametric test

In 1970, a lottery was used to determine who would be drafted into the U.S. Army. The 366 dates in the year were placed in individual capsules. First , the 31 Jan. capsules were placed in a box; then the 29 Feb. capsules were added and the two months were mixed. Then the 31 March capsules were added and the three months were mixed. This process continued until all months were included. The first capsule selected was Sept. 14, so men born on that date were drafted first. The accompanying spreadsheet shows the 366 dates in the order of selection.

ANALYZING THE RESULTS

a. Use the runs test to test the sequence for randomness above and below the median of 183.5.

b. Use the Kruskal-Wallis test to test the claim that the 12 months had priority numbers drawn from the same population.

c. Calculate the 12 monthly means. (The horizontal scale list 12 months, and the vertical scale ranges from 100 to 260) Note any pattern suggesting that the original priority numbers were not randomly selected.

d. Based on the results from a, b, and c, was this lottery fair?

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#### Solution Preview

Please see attached file.

Runs Test

The runs test can be used to decide if a data set is from a random process. A run is defined as a series of increasing values or a series of decreasing values. The runs test is a non-parametric test that checks the randomness hypothesis of a data sequence. The Runs Test procedure tests whether the order of occurrence of two values of a variable is random. A run is a sequence of like observations. A sample with too many or too few runs suggests that the sample is not random. You can use the Runs Test procedure to test whether the order of values of a variable is random. The procedure first classifies each value of the variable as falling above or below a cut point and then tests to ensure that there is no order to the resulting sequence.

The test indicates a pattern of 183 above the median of 183.5 and 183 below with G = 174 (number of runs). The sequence of data appears to have the same characteristic and the number of runs is not too high or too low which indicates a acceptance for randomness. Here the significance value (P value) is greater than 0.05. Thus we accept null hypothesis that the data is random. A monthly analysis also leads to the same conclusion

Runs Test

Statistic

Test Value 183.50

Total Cases 366

Number of Runs 174

Z -1.047

Asymp. Sig. (2-tailed) .295

Runs Test : Month wise

Test Value = 183.5

Month Total

Cases Number of

Runs Z Asymp. ...

#### Solution Summary

The solution gives the details of Kruskal Wallis nonparametric test to examine the randomness of lottery data.