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

## 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.

### 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.

### Test Procedure of sign test

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.

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.

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