The Sign Test

The simplest nonparametric test to carry out is the sign test. It is used for testing hypotheses about the central location of a population distribution and is most frequently employed in analyzing data from matched pairs.

Table 10.1 shows the results of a taste comparison experiment. A manufacturer of baked beans was contemplating a new recipe for the sauce used in its product. A random sample of eight individuals was chosen, and each was asked to rate on a scale from 1 to 10 the taste of the original and the proposed new product. The scores are shown in the table. Also shown are the differences in the scores for every taster and the signs of these differences. Thus, a + is assigned if the original product is preferred, a — if the new product is preferred, and 0 if the two products are rated equally. In this particular experiment, two tasters preferred the original product and five the new; one rated them equal.

The null hypothesis of interest is that in the population at large, there is no overall tendency to prefer one product over the other. In assessing this hypothesis, we compare the numbers expressing a preference for each product, discarding those who rated the products equally. In the present example then, the effective sample size is reduced to 7, and the only sample information on which our test is based is that two of the seven tasters preferred the original product.

The null hypothesis can be viewed as the hypothesis that the population median of the differences is 0 (which would be true, for example, if the differences came from a population whose distribution was symmetric about a mean of 0). If this hypothesis were true, our sequence of + and — differences could be regarded as a random sample from a population in which the probabilities for -I- and - were each .5. In that case, the observations would constitute a random sample from a binomial population in which the probability of + was .5. Thus, if p denotes the true proportion of+'s in the population, the null hypothesis is simply

We may want to test this hypothesis against either a one-sided or a two-sided alternative. Suppose that in the taste preference example, the alternative of interest is 