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)
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(a) To calculate the probability of a Type I error, we need to understand that a Type I error occurs when we reject the null hypothesis (that the applicant has no taste sensitivity) when it is actually true. In this case, the null hypothesis states that the applicant correctly identifies the odd sample at most 5 times out of the 10 trials. So, the probability of a Type I error is the probability of observing 6, 7, 8, 9, or 10 correct identifications out of 10 trials, given that the null hypothesis is true.
To calculate this probability, we need to find the cumulative probability of observing 6, 7, 8, 9, or 10 successes in a binomial distribution with n=10 trials and p=1/3 (the probability of success under the null hypothesis). Using a statistical software or table, we find this cumulative probability to be approximately 0.212.
Therefore, the probability of a Type I error is approximately 0.212 or 21.2%.
(b) To calculate the probability of a Type II error, we need to understand that a Type II error occurs when we fail to reject the null hypothesis (that the applicant has no taste sensitivity) when it is actually false. In this case, the null hypothesis states that the applicant correctly identifies the odd sample at most 5 times out of the 10 trials, whereas the alternative hypothesis states that the applicant has taste discrimination ability, with a probability of identifying the odd sample equal to p=0.5.
To calculate this probability, we need to find the cumulative probability of observing 0, 1, 2, 3, 4, or 5 successes in a binomial distribution with n=10 trials and p=0.5 (the probability of success under the alternative hypothesis). Using a statistical software or table, we find this cumulative probability to be approximately 0.376.
Therefore, the probability of a Type II error is approximately 0.376 or 37.6%.
(c) To calculate the probability of a Type I error with a rejection region of 8, 9, or 10, we use the same reasoning as in part (a). We need to find the cumulative probability of observing 8, 9, or 10 successes in a binomial distribution with n=10 trials and p=1/3 (the probability of success under the null hypothesis).
Using a statistical software or table, we find this cumulative probability to be approximately 0.065.
Therefore, the probability of a Type I error with a rejection region of 8, 9, or 10 is approximately 0.065 or 6.5%.
(d) To calculate the probability of a Type II error with a rejection region of 8, 9, or 10, we use the same reasoning as in part (b). We need to find the cumulative probability of observing 0, 1, 2, 3, 4, 5, 6, or 7 successes in a binomial distribution with n=10 trials and p=0.5 (the probability of success under the alternative hypothesis).
Using a statistical software or table, we find this cumulative probability to be approximately 0.122.
Therefore, the probability of a Type II error with a rejection region of 8, 9, or 10 is approximately 0.122 or 12.2%.
(e) To recommend a rejection region to the brewing company, we need to consider the trade-off between Type I and Type II errors. A Type I error occurs when we mistakenly reject the null hypothesis (that the applicant has no taste sensitivity) when it is actually true. On the other hand, a Type II error occurs when we fail to reject the null hypothesis when it is actually false.
In this case, the brewing company wants to be relatively confident in their decision to hire an applicant with taste discrimination ability. They want to minimize the risk of hiring an applicant who lacks taste sensitivity. Therefore, they should choose a rejection region that minimizes the probability of a Type I error.
Comparing the rejection regions given, the rejection region of 8, 9, or 10 has the lowest probability of Type I error (approximately 6.5%). This means that if an applicant correctly identifies the odd sample at least 8 times out of 10 trials, the brewing company can be relatively confident in their decision to hire them as a taster.
Therefore, I would recommend that the brewing company use the rejection region of 8, 9, or 10.