17.5.1.2 Algorithms (One-Sample Wilcoxon Signed Rank Test)

The Wilcoxon Sign Rank test is used to replace the one sample t-test, when the normality is questionable. So it requires looser conditions than the one sample t-test and has a wider use than the t-test.

a) For each $x_i\,\!$ , for $i=1,2,\ldots ,n$ , the signed difference $d_i=x_i-\mu _0\,\!$ is found,where $\mu _0\,\!$ is a given test value for the median of the sample.

b) Ignore the cases where $d_i=0\,\!$ . Rank the rest of $\left| d_i\right|$ , use $r_i$ as its rank. Pay attention to that any tied values of $\left| d_i\right|$ are assigned the average of the tied ranks. For example, three? $\left| d_i\right|$ ranked as 7 8 9 are ties, then their rank is (7+8+9) /3=8.?

c) To each rank is affixed the sign of $d_i\,\!$ to which it corresponds. Let $s_i=sign(d_i)r_i\,\!$

d) The sum of the positive-signed ranks is calculated as

$W_1=\sum_{s_i>0}s_i$

Our null hypothesis is that the population median has a specific value $\mu _0\,\!$ . We test the null hypothesis against the two-sided alternative hypothesis that the population does not have a median value $\mu _0\,\!$ . The confidence interval is converted to hypothesis-test form. The test is a one-sample Wilcoxon Sign Rank test, and it is defined as:

$H_0$	$\mu =\mu _0\,\!$
$H_1$	$\mu \neq \mu _0$
Test Statistic	$z=\frac{(W-\frac{n_1(n_1+1)}4)-\frac 12\cdot sign(W-\frac{n_1(n_1+1)}4)}{\sqrt{\frac 14\cdot\sum_{i=1}^n S_i^2}}$ Where $W\,\!$ , $s_i\,\!$ is said above $n_1\,\!$ , and is the number of non-zero $d_i\,\!$ , .
Significance Level $\alpha \,\!$ :	The most commonly used value for $\alpha \,\!$ is 0.05.
Critical Region:	Reject the null hypothesis that the median is a specified value, $\mu _0\,\!$ , if $\left\| z\right\| >Z_{\alpha /2}$ ,where Z~N(0,1) Because for large sample, for example the size of the population is more than 50, sized the distribution of is approximately standard normal.