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 someway 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.

Question

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 someway 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? _______________ (3 pts)

(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? _______________ (3 pts)

(c) With a Rejection Region of 8, 9 or 10, what is the probability of a Type I error? _______________ (3 pts)

(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? __________ (3 points)

(e) Which Rejection Region would you recommend that the brewing company use? Why? (1 pt)

Answer 1

To calculate the probabilities of Type I and Type II errors and recommend the appropriate Rejection Region, we need to use statistical hypothesis testing. Let's break down each question step by step:

(a) The probability of a Type I error, denoted as alpha (α), is the probability of rejecting the null hypothesis when it is actually true. In this case, the null hypothesis is that the applicant has no ability to discern differences in beer samples.
Since the Rejection Region is defined as 6, 7, 8, 9, or 10, this means if the applicant correctly identifies the odd sample at least 6 times out of 10 trials, we reject the null hypothesis. To calculate the probability of a Type I error, we need to assume that the null hypothesis is true and find the probability of observing a test statistic (number of correct identifications) in the Rejection Region.
To calculate this probability, we can use a binomial distribution. The probability of getting exactly k correct identifications out of 10 trials can be calculated using the formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials (10 in this case) and p is 1/3 (probability of a correct identification by chance).
We need to calculate the sum of probabilities for k=6, 7, 8, 9, and 10:
P(Type I error) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

(b) The probability of a Type II error, denoted as beta (β), is the probability of failing to reject the null hypothesis when it is actually false. In this case, the null hypothesis is that the applicant has no taste discrimination ability (p = 0.5).
To calculate the probability of a Type II error, we again assume the null hypothesis is false. We need to find the probability of observing a test statistic (number of correct identifications) that falls outside the Rejection Region defined as 6, 7, 8, 9, or 10.
Using a similar approach as in part (a), we calculate the sum of probabilities for k=0, 1, 2, 3, 4, 5:
P(Type II error) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

(c) With a Rejection Region of 8, 9, or 10, the calculation is similar to part (a). We need to calculate the sum of probabilities for k=8, 9, and 10. Using the same binomial distribution formula, we can find:
P(Type I error) = P(X=8) + P(X=9) + P(X=10)

(d) With the same Rejection Region as part (c), we need to calculate the sum of probabilities for k=0, 1, 2, 3, 4, 5, 6, and 7, using the binomial distribution formula:
P(Type II error) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7)

(e) To recommend the appropriate Rejection Region, we need to consider the importance of Type I and Type II errors and the company's goals.
If Type I errors are more costly (such as erroneously hiring someone who cannot discern taste differences), the company might want to have a lower probability (α) and choose a more stringent Rejection Region, like 8, 9, or 10.
If Type II errors are more costly (such as failing to hire someone with good taste discrimination ability), the company might want to have a lower probability (β) and choose a less stringent Rejection Region, like 6, 7, 8, 9, or 10.
Ultimately, the recommended Rejection Region depends on the company's preferences and the trade-off between the two types of errors.

To obtain the exact values for probabilities, you need to perform the binomial calculations described above using a statistical software or calculator that can evaluate binomial probabilities.