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)
(a) To calculate the probability of a Type I error, we need to find the probability of rejecting the null hypothesis when it is actually true. In this case, the null hypothesis is that the applicant has no taste sensitivity.
Since the applicant's probability of identifying the odd sample is not given, we assume it to be 0.5 (p = 0.5).
The probability of correctly identifying the odd sample at least 6 times out of 10 trials can be calculated using the binomial probability formula: P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).
Using a binomial probability calculator or a statistical software, we can find the probability of each individual term in the above equation and sum them up to get the probability of a Type I error.
(b) To calculate the probability of a Type II error, we need to find the probability of failing to reject the null hypothesis when it is actually false. In this case, the alternative hypothesis is that the applicant has taste discrimination ability, with a probability of identifying the odd sample of p = 0.5.
The probability of failing to identify the odd sample at least 6 times out of 10 trials can be calculated using the binomial probability formula: P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).
Using a binomial probability calculator or a statistical software, we can find the probability of each individual term in the above equation and sum them up to get the probability of a Type II error.
(c) With a Rejection Region of 8, 9, or 10, the probability of a Type I error can be calculated in the same way as in part (a). Calculate and sum the probabilities of P(X = 8), P(X = 9), and P(X = 10) using the binomial probability formula.
(d) With a Rejection Region of 8, 9, or 10, the probability of a Type II error can be calculated in the same way as in part (b). Calculate and sum the probabilities of P(X = 0), P(X = 1), P(X = 2), P(X = 3), P(X = 4), P(X = 5), P(X = 6), and P(X = 7) using the binomial probability formula.
(e) To recommend a rejection region, we need to consider the trade-off between Type I and Type II errors. A smaller rejection region (fewer correct identifications required) increases the chance of a Type I error (rejecting the null hypothesis when it is true), while a larger rejection region decreases the chance of a Type I error but increases the chance of a Type II error (failing to reject the null hypothesis when it is false).
Depending on the importance of these errors to the brewing company, we can recommend a rejection region based on their priorities. If the company wants to be more cautious and avoid mistakenly hiring someone without taste sensitivity (Type I error), they may choose a smaller rejection region like the one with 8, 9, or 10 correct identifications. On the other hand, if the company wants to ensure they don't miss out on potential candidates with taste discrimination ability (Type II error), they may choose a larger rejection region like the one with 6, 7, 8, 9, or 10 correct identifications.
Ultimately, the choice of the rejection region depends on the specific priorities and requirements of the brewing company.