A common application of the sign test deals with analyzing consumer preferences. For instance, suppose that a blind taste test is administered to nine randomly selected convenience store customers. Each participant is asked to express a preference for either Coke or Pepsi after tasting unidentified samples of each soft drink. The sample results are expressed by recording a +1 for each consumer who prefers Coke and a - 1 for each consumer who prefers Pepsi. Note that sometimes, rather than recording either a +1 or a - 1, we simply record the sign + or -, hence the name "sign test." A 0 is recorded if a consumer is unable to rank the two brands, and these observations are eliminated from the analysis.
The null hypothesis in this application says that there is no difference in preferences for Coke and Pepsi. If this null hypothesis is true, then the number of + 1 values in the population of all preferences should equal the number of - 1 values, which implies that the median preference Md = 0 (and that the proportion p of + 1 values equals .5). The alternative hypothesis says that there is a significant difference in preferences (or that there is a significant difference in the number of +1 values and - 1 values in the population of all preferences). This implies that the median preference does not equal 0 (and that the proportion p of + 1 values does not equal .5).
a. Table 18.1 gives the results of the taste test administered to the nine randomly selected consumers. If we consider testing H0: Md = 0 versus Ha: Md ≠ 0 where Md is the median of the (+1 and - 1) preference rankings, determine the values of S1, S2, and S for the sign test needed to test H0 versus Ha. Identify the value of S on the Excel add-in (MegaStat) output,
b. Use the value of S to find the p-value for testing H0: Md = 0 versus Ha: Md ≠ 0. Then use the p-value to test H0 versus Ha by setting a equal to .10, .05, .01, and .001. How much evidence is there of a difference in the preferences for Coke and Pepsi? What do you conclude?