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

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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. According to wikipedia, “The sign test is a statistical method to test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment.”

 

Sign Test

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 

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

Hypothesis

We want to test the following hypothesis.

  hypothesis of sign test

Test Procedure

Since, the population is symmetric about μ. Then the probability of sample value exceeding the mean μ and the probability of a sample value less than μ 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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