Sign Test: Step by Step calculations for small and large sample case



Sign Test by statisticalaid.com


Sign Test

The sign test is the oldest of all non-parametric tests. and it is so called as usually covert the data for analysis to a series of plus and minus sign. The test statistic consists of either the number of plus sign or the number of minus sign. 

Let, x1,x2,......... xn be a random sample of size n from a N population with mean μ.

Sign test for small sample case (n< 35)   

Assumptions of sign test

  •                The population is continuous and symmetric about mean.
  •                xi’s  are independent.


Hypothesis of sign test

We want to test the following hypothesis.

1.    hypothesis of sign test


 
Test Procedure of sign test

Since, the population is symmetric about Î¼. Then the probability of sample value exceeding the mean Î¼ and the probability of a sample value less than Î¼ are both 1/2 , get the difference, (xi-μ0); i=1,2,...,n. Put (t) sign if  (xi-μ0) > 0 and minus sign if  (xi-μ0) < 0 for some i. If,  (xi-μ0)=0 then  simply discard it from the analysis and n is reduced accordingly. Note that under H0, the number of plus sign is binomial variance with index n and parameter θ=1/2.

test procedure of sign test


Conclusion: If the observed number of plus sign falls in the critical region, we may reject our Ho. otherwise we may accept ho.

 

Sign test for large sample (n>35) 

 

For large sample H0 may be test by,

Where, z is approximately normally distributed with mean 0 and variance 1. The approximation to the normal distribution becomes better if a correction for continuity is used. The correction is necessary since the normal distribution is continuous while the normal distribution is involved  discrete variable. The corrected 2 becomes.

sign test for large sample


Comment: If the calculated value of  z falls in the critical region then we may reject our H0 otherwise accept our H0.

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