/math-d2583020b138319a535bc3c88278ab33.png?v=0 "x_i\,\!")
, for/math-6b80d670336d4b43b32dbd006770a4f1.png?v=0 "i=1,2,\ldots ,n")
, the signed difference/math-0816d4a592c11fc9915d60c0ab5c8ba8.png?v=0 "d_i=x_i-\mu _0\,\!")
is found,where /math-4ed1ed38631c7263bc6b664a5c510f8e.png?v=0 "\mu _0\,\!")
is a given test value for the median of the sample.
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 , for , the signed difference is found,where is a given test value for the median of the sample.
b) Ignore the cases where /math-a3bf95053be7ee784232acceb28b8046.png?v=0 "d_i=0\,\!")
. Rank the rest of/math-80c599954d244f780dfa6152785f1d15.png?v=0 "\left| d_i\right|")
, use/math-af2593b49488f8bc44396795792c2d79.png?v=0 "r_i")
as its rank. Pay attention to that any tied values of are assigned the average of the tied ranks. For example, three? 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 /math-40b4081d7e04c18f0529a2e70a11e290.png?v=0 "d_i\,\!")
to which it corresponds. Let /math-1e45800d815894820f5acad906bbb8e7.png?v=0 "s_i=sign(d_i)r_i\,\!")
d) The sum of the positive-signed ranks is calculated as
/math-ead8d10bd1463b7b0bd40d8977d5ba37.png?v=0 "W_1=\sum_{s_i>0}s_i")
Our null hypothesis is that the population median has a specific value . We test the null hypothesis against the two-sided alternative hypothesis that the population does not have a median value . The confidence interval is converted to hypothesis-test form. The test is a one-sample Wilcoxon Sign Rank test, and it is defined as:
/math-e65765bedcabe42c66ec93228769e82a.png?v=0 "H_0")
|
/math-0a9a0062a3140e27523057ac492950e3.png?v=0 "\mu =\mu _0\,\!")
|
|---|---|
/math-6207a80403dcccc1aa3b5b7303315c4b.png?v=0 "H_1")
|
/math-858de1a56ceb15609e7ab979a2638b00.png?v=0 "\mu \neq \mu _0")
|
| Test Statistic | /math-b1b86035c067633cdfc2a356665fc977.png?v=0 "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 |
Significance Level /math-4bc6c42bbabe567d1f2516326e52b775.png?v=0 "\alpha \,\!") :
|
The most commonly used value for is 0.05.
|
| Critical Region: | Reject the null hypothesis that the median is a specified value, , if
Because for large sample, for example the size of the population is more than 50, sized the distribution of is approximately standard normal. |
For more details of the algorithm, please refer to nag_wilcoxon_test (g08agc)
/math-e5631ee132b76058c37eb3abeed99259.png?v=0 "W\,\!")
,/math-df637c345145e684f09c3e2bf73732cf.png?v=0 "s_i\,\!")
is said above /math-8d88a43872d6db6535d8672a15f09ce2.png?v=0 "n_1\,\!")
, and is the number of non-zero /math-49f05dc56aa54a13a63962746c7fff75.png?v=0 "\left| z\right| >Z_{\alpha /2}")
,where Z~N(0,1)