Recall from our discussion in class (also see the website) that the hypothesis testing framework can be applied by food and beverage companies seeking to hire "tasters" who have above average taste-discrimination ability. Suppose a brewer intends to use the triangle taste test method to identify the best applicants for the position of "taster." (It does so because the company will want their tasters to have sensitive palates, and it will need some way of determining beforehand whether a candidate for a taster's job can detect subtle differences in taste.) Remember how this works: in a single trial of this test, the applicant is presented with 3 samples of beer---two of which are alike---and is asked to identify the odd sample. Except for the taste difference, the samples are as alike as possible (same color, same temperature, same cup, and so on). To check the applicant's ability, he/she is presented with a series of triangle tests. The order of the presentation is randomized within each trial. Clearly, in the absence of any ability at all to distinguish tastes, the probability the applicant will correctly identify the odd sample in a single trial is one-third (i.e., 1/3). The question---and the question the brewing company wants to answer---is whether the applicant can do better than this. More specifically, if an applicant with no ability to distinguish tastes is presented with 12 trials, we would expect that he/she would correctly identify the odd sample 4 times, simply by luck alone. On the other hand, if an applicant with a sensitive palate is presented with 12 trials, we wouldn't be surprised to see him/her correctly identify the odd sample more frequently than this, and perhaps much more frequently.
Suppose our null hypothesis is that a certain applicant has no ability at all to discern differences in beer samples, and that to her one beer tastes pretty much the same as another. Obviously, the alternative hypothesis states that the applicant does have taste discrimination ability. Our job is to present this person with a series of triangle taste tests with the purpose of collecting data (the number of correct identifications made in a series of trials) which help the brewing company classify the job applicant into one of the two groups.
If the company administers n = 10 identical triangle taste tests to this job applicant and if we say that 'x' is the number of correct identifications made (in n=10 trials), then the Rejection Region is the "set of values which 'x' could assume that will lead us to reject the null hypothesis, and prefer the alternative hypothesis."
We could choose any Rejection Region we like, but suppose the company decides it should be: 6, 7, 8, 9, or 10. That is, if after an applicant is presented with n=10 triangular taste tests (or 10 trials), she correctly identifies the odd sample at least 6 times, we reject the null hypothesis (that the applicant has no taste sensitivity) and prefer the alternative hypothesis (that the applicant has taste discrimination ability), and we make her an offer of employment as a taster.
(a) With a Rejection Region of 6, 7, 8, 9 or 10, what is the probability of a Type I error?
(b) With a Rejection Region of 6, 7, 8, 9 or 10, what is the probability of a Type II error, if the job applicant has a probability of identifying the odd sample with p = 0.5?
(c) With a Rejection Region of 8, 9 or 10, what is the probability of a Type I error?
(d) With a Rejection Region of 8, 9 or 10, what is the probability of a Type II error, if the job applicant has a probability of identifying the odd sample with p = 0.5?
(e) Which Rejection Region would you recommend that the brewing company use? Why?