Sign Test

Sign Test

Consider the set of paired observations (X₁, Y₁), (X₂, Y₂), ...., (X_n, Y_n).
Let p = P(Y_i > X_i). To find out if the values of the Y_i's are probabilistically higher than its paired X_i value, we would design a sample test with the null hypothesis

H_o : p = 0.5

The null hypothesis states that the values of the Y_i's have the same probability of being higher than the corresponding X_i value as they do being lower than the X_i's. On the other hand, the alternative hypothesis states that the values of the Y_i's have a tendency to be higher (or lower) than the corresponding X_i value.

For each (Y_i, X_i) observation, define the random variable B_i, where

Let B = (B₁ + B₂ + .... + B_n). Then the set of random variables B₁, B₂, ...., B_n vare independent Bernoulli variables with parameter p, and the random variable B is binomially distributed with parameters p and n. Therefore, if the null hypothesis is true, then B would have a binomial distribution with parameters p = 0.5 and n.

For large n, the test statistic

would have an approximately standard normal distribution (due to the central limit theorem). The following table shows the critical region of this sample test for various alternative hypothesis, using an a level of significance: