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A histogram of white blood cell (WBC) counts in 15 sick patients showed that the distribution was negatively skewed. If we wanted to test for differences between the published WBC count for a healthy population compared to the WBC values in these patients which type of test should be used?

Answer is saying Wilcoxon signed rank test rather than Mann-Whitney and i'm not sure why?

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  • $\begingroup$ Welcome to Cross Validated! You have a single sample, don't you? Please describe your data in more detail. $\endgroup$
    – Dave
    Nov 17, 2020 at 21:40
  • $\begingroup$ I thought the Mann–Whitney U test is applied to independent samples while the Wilcoxon signed-rank test is applied to matched paired samples. So it depends on your data $\endgroup$
    – Henry
    Nov 17, 2020 at 21:43
  • $\begingroup$ Hi, thanks for replies. It’s a MCQ from my course and I’m assuming they are asking to compare the sick patients with a healthy sample so it’s independent rather than matched pairs? So I can’t initially see why it isn’t Mann Whitney as they sick patients and the healthy sample aren’t related presumably $\endgroup$
    – Geoff
    Nov 17, 2020 at 21:50

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Just realised the answer. It’s because the second group isn’t actually a group it’s a published average for the population. So it’s the non parametric version of the one sample t test rather than independent t test.

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  • $\begingroup$ This the reason it isn't a two-sample Wilcoxon test. But you could use a one-sample Wilcoxon test (which isn't exclusively for paired data). $\endgroup$
    – BruceET
    Nov 18, 2020 at 21:38
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I have no idea what values you may have for WBC, but here is a left-skewed sample of size $n = 15$ with median around $H = 17.$ If the sample in your publication has population median around $\eta = 14,$ then your subjects are from a different population.

set.seed(2020)
x = 20*rbeta(15, 10, 2)
boxplot(x, horizontal=T)

enter image description here

summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  8.236  14.891  16.981  16.193  18.584  19.929 

wilcox.test(x, mu=14)

        Wilcoxon signed rank test

data:  x
V = 100, p-value = 0.02155
alternative hypothesis: true location is not equal to 14

The difference between 17 and 14 is relatively large. With only $n= 15$ observations you may not have very good power (ability to detect a real difference). For example:

wilcox.test(x, mu=15)$p.val
[1] 0.1069946
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