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## Biostatistics: Wilcoxon Rank Sum Test

Biostatistics: Wilcoxon Rank Sum Test

1. CHAPTER 13, PROBLEM 11

#11.  The characteristics of low birth weight children dying of sudden infant death syndrome were examined for both females and males. The ages at time of death for samples of 11 girls and 16 boys are shown below.

Age (days)

 Females Rank Males Rank 53 3 46 1 56 4 52 2 60 7.5 58 5 60 7.5 59 6 78 10.5 77 9 87 15 78 10.5 102 16 80 12 117 20 81 13 134 22.5 84 14 160 24.0 103 17 277 27 114 18 115 19 133 21 134 22.5 167 25 175 26 157 221

a) Test the null hypothesis that the median ages at death are identical for the two populations.  What do you conclude?

HYPOTHESIS
– Hypotheses in terms of the medians of the two distributions (populations)
– Ho: Medianpop1 = Medianpop2
– Ha: Medianpop1 ≠ Medianpop2

SAMPLE INFORMATION
– Independent sample #1:
– Size n1 = 11
– (Let size n1, or sample 1 be the one with the samller sample size)

– Independent sample #2:
– Size n2 = 16

COMBINE THE DATA
THE SUMS
– W = sum of ranks in the smaller group (sum of n1 ranks)
– What is W? W = 157

TO TO TABLE A.7
– n2 = 16; – Find the table for your n2.
– n1 = 11; – Find the column for n1.
– What is your W? 157

– Pr(W≤W) = the area you see.
– What is Pr(W≤W)?

MEAN = E(W) = Uw = [n1(n1+n2+1)]/2
Uw = [11(11+16+1)]/2
Uw = 154

VARIANCE = Var(W) = σ2w = [n1n2(n1+n2+1)]/12
Var(W) = σ2w = [11X16(11+16+1)]/12
Var(W) = σ2w = [11X16(11+16+1)]/12
Var(W) = σ2w = 410.6667
SD= σw = 20.264

COMPUTE THE Z STATISTIC
–  z = [W – Uw]/ σw

z = {W – [n1(n1+n2+1)/2]} / sqrt{[n1n2(n1+n2+1)]/12}
z = [157 – 154]/ 20.264
z = 0.148 = 0.15

Find Pr(Z<z) using Table A.3 (N(0,1) distribution)
Pr(Z<z) = 0.440

CALCULATE THE P VALUE
– For a two sided test, double that probability
p < 0.880

I conclude that there is no difference between the median ages at death for males and females.

b) Do you feel that it would have been appropriate to use the two-sample t-test to analyze these data? Why or why not?

I do not feel that a two-sample t-test would be appropriate.  The data for males and females seems to be skewed to the right in terms of age of death, making it inappropriate to use the two sample t-test.

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