Suppose we are interested in comparing preferences for two products, such as Diet Coke and Diet Pepsi. We collect a 7-point scale satisfaction rating for each drink from a sample of 100 college students. We would like to know if there is statistical evidence that there is a preference difference between the two drinks. Would we apply a one-tailed or two-tailed test?
Here we should apply a two-tailed test because we are interested in a difference in either direction. Our null hypothesis is that the two population means are equal, H0: μ1 = μ2 while the alternative hypothesis is that the two population means are not equal, H1: μ1 ≠ μ2.
If we decide to use alpha = .05, then we would reject H0 if we find a difference between means in either direction that has a p-value of .025 or less in the tail. If the null hypothesis is true, the probability that we would observe such a great difference between sample means is .025 + .025 = .050.