Another cunning application for binomial tests arises when we have two measurements for each subject in a study, and we want to test whether one of the measurements is systematically larger than the other. For this sort of analysis we would generally use a test called the paired t-test. However, if we aren’t interested in the size of the differences, and only want to know whether one measurement is systematically larger, we can use the simple binomial test instead.

Earthquake magnitudes can be measured on two different scales: the Richter scale, and the moment scale. The moment magnitude scale is considered more accurate, but the Richter scale is still in common use. Table 33.1 gives both measurements for major earthquakes in New Zealand from 2010 to 2020, with columns R and M denoting the measurements on the Richter and moment scales respectively.

Using the data in Table 33.1, score each earthquake as 1, 0, or

−1 according to whether the Richter measurement is higher, equal, or lower than the moment measurement. Design suitable hypotheses to test whether the Richter scale gives readings systematically higher or lower than the moment scale. Carry out your test and interpret your findings. (Hint: you can ignore the earthquakes with equal scores.)