Lady tasting tea

The experiment

The experiment provides a subject with eight randomly ordered cups of tea – four prepared by pouring milk and then tea, four by pouring tea and then milk. The subject attempts to select the four cups prepared by one method or the other, and may compare cups directly against each other as desired. The method employed in the experiment is fully disclosed to the subject.

The null hypothesis is that the subject has no ability to distinguish the teas. In Fisher's approach, there was no alternative hypothesis, unlike in the Neyman–Pearson approach.

The test statistic is a simple count of the number of successful attempts to select the four cups prepared by a given method. The distribution of possible numbers of successes, assuming the null hypothesis is true, can be computed using the number of combinations. Using the combination formula, with n=8 total cups and k=4 cups chosen, there are \binom{8}{4} = \frac{8!}{4!(8-4)!} = 70 possible combinations.

The frequencies of the possible numbers of successes, given in the final column of this table, are derived as follows. For 0 successes, there is clearly only one set of four choices (namely, choosing all four incorrect cups) giving this result. For one success and three failures, there are four correct cups of which one is selected, which by the combination formula can occur in \binom 41 =4 different ways (as shown in column 2, with x denoting a correct cup that is chosen and o denoting a correct cup that is not chosen); and independently of that, there are four incorrect cups of which three are selected, which can occur in \binom 43 = 4 ways (as shown in the second column, this time with x interpreted as an incorrect cup which is not chosen, and o indicating an incorrect cup which is chosen). Thus a selection of any one correct cup and any three incorrect cups can occur in any of 4×4 = 16 ways. The frequencies of the other possible numbers of successes are calculated correspondingly. Thus the number of successes is distributed according to the hypergeometric distribution. Specifically, for a random variable X equal to the number of successes, we may write X \sim \operatorname{Hypergeometric}(N=8,K=4,n=4), where N is the population size or total number of cups of tea, K is the number of success states in the population or four cups of either type, and n is the number of draws, or four cups. The distribution of combinations for making k selections out of the 2k available selections corresponds to the kth row of Pascal's triangle, such that each integer in the row is squared. In this case, k = 4 because 4 teacups are selected from the 8 available teacups.

The critical region for rejection of the null of no ability to distinguish was the single case of 4 successes of 4 possible, based on the conventional probability criterion 5%).

Thus, if and only if the lady properly categorized all 8 cups was Fisher willing to reject the null hypothesis – effectively acknowledging the lady's ability at a 1.4% significance level (but without quantifying her ability). Fisher later discussed the benefits of more trials and repeated tests.

David Salsburg reports that a colleague of Fisher, H. Fairfield Smith, revealed that in the actual experiment the lady succeeded in identifying all eight cups correctly. The chance of someone who just guesses of getting all correct, assuming she guesses that any four had the tea put in first and the other four the milk, would be only 1 in 70 (the combinations of 8 taken 4 at a time).

''The Lady Tasting Tea'' book

David Salsburg published a popular science book entitled The Lady Tasting Tea, which describes Fisher's experiment and ideas on randomization. Deb Basu wrote that "the famous case of the 'lady tasting tea was "one of the two supporting pillars ... of the randomization analysis of experimental data."

References

• {{cite journal |author-link=Debabrata Basu
• Basu, D. (1980b). "The Fisher Randomization Test", reprinted with a new preface in Statistical Information and Likelihood : A Collection of Critical Essays by Dr. D. Basu; J. K. Ghosh, editor. Springer 1988.
• {{cite book| last=Kempthorne|first=Oscar|author-link=Oscar Kempthorne|chapter=Intervention experiments, randomization and inference |chapter-url=http://projecteuclid.org/euclid.lnms/1215458836
• Salsburg, D. (2002) The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century, W.H. Freeman / Owl Book.

References

1. OED quote: '''1935''' R. A. Fisher, ''[[The Design of Experiments]]'' ii. 19, "We may speak of this hypothesis as the 'null hypothesis' [...] the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation."
2. Salsburg (2002)
3. Basu (1980a, p. 575; 1980b)