Sign Test for Small samples





Why Sign test?





In statistics we usually deals with parametric tests. To conduct a parametric test, certain assumptions have to be satisfied. Such as the normality assumption. But sometimes, these assumptions are violated. In that case we will use the corresponding non-parametric alternatives.





Actually non parametric tests can be used every time instead of parametric tests. But due to the popularity of the parametric tests, we usually use non-parametric tests, when the assumptions of the parametric tests are violated.





Sign test is the non-parametric alternative for the one sample T test. In one sample T test, we test for the population mean. But in sign test, we test for the population median.





When to use the Sign test?

When the sample size is large (greater than 30), we can use the one sample t test. Because using the central limit theorem we can say that the distribution of the sample mean is approximately normally distributed. To use the one sample T test when the sample size is small, the corresponding population should be normally distributed. But what if the population that the sample was taken is not normally distributed?





In this situation, we can use the sign test as the assumptions of the one sample T test has violated.





Consider the following example. The monthly income of 6 randomly chosen students are $1200, $750, $1250, $950, $1050 and $1450 respectively. Is there evidence that the median income of students is more than $950?





Following steps should follow in sign test





Step 1 : Identify the null and alternative hypothesis









Step 2: Calculate the test statistic.





The test statistic C is the number of values (+ values) greater than median. To do this, first arrange the data in either ascending or descending order. So in this example the ascending order is $750, $950, $1050, $1200, $1250 and $1450.





The values which are greater than median are + values. And the values which are less than median are – values. The values that are exactly equal to the median should be ignored. The sample size n* should be the all values excluding the number of observations which are equal to the median. . The sample size nshould be the all values excluding the number of observations which are equal to the median.





* = 5 and C = 4 So in this example, n= 5 and C = 4





The test statistic C has the following distribution under the null hypothesis.





*, 0.5) C ~ Bin (n, 0.5)





Step 2: Calculate the P value.





P value = P(C >=4) .

So P value = P(C=4) + P(C=5). To calculate this you can use binomial table or manually calculate each probability.









So p value is =0.1875.





So the P value is greater than 0.05. Therefore the null hypothesis is not rejected.





This is how we can conduct sign test manually. In next post we will discuss how to do this using minitab.







