Chapter 9: Hypothesis Tests: Introduction, Basic Concepts, and an Example

1. With a Rejection Region of RR8 = {8, 9, 10, 11, 12, 13, 14}, what is the probability of a Type I error? Recall that since a Type I error occurs when the subject has no taste-discrimination ability, p = 1/3.

Answer: 0.0576. This is equal to the probability of 8 or more correct identifications if the subject has no taste discrimination ability (and p = 1/3)
1 - pbinom(7, 14, 1/3)
## [1] 0.0576163
#or
sum(dbinom(8 : 14, 14, 1/3))
## [1] 0.0576163

2. With a Rejection Region of RR8 = {8, 9, 10, 11, 12, 13, 14}, what is the probability of a Type II error, if the subject has a probability of p = 0.80 of identifying the odd sample?

Answer: 0.0116. That is, the probability of 7 or fewer correct identifications if the subject is able to identify the odd sample with 0.80 probability.

pbinom(7, 14, 0.80)
## [1] 0.01160991
#or
sum(dbinom(0 : 7, 14, 0.80))
## [1] 0.01160991

3. With a Rejection Region of RR10 = {10, 11, 12, 13, 14}, what is the probability of a Type I error?

Answer: 0.0040. This is equal to the probability of 10 or more correct identifications if the subject has no taste discrimination ability (and p = 1/3)
1 - pbinom(9, 14, 1/3)
## [1] 0.004039541
#or
sum(dbinom(10 : 14, 14, 1/3))
## [1] 0.004039541

4. With a Rejection Region of RR10 = {10, 11, 12, 13, 14}, what is the probability of a Type II error, if the subject has a probability of p = 0.80 of identifying the odd sample?

Answer: 0.1298. This is equal to the probability of 9 or fewer correct identifications if the subject is able to identify the odd sample with 0.80 probability.
pbinom(9, 14, 0.80)
## [1] 0.1298396
#or
sum(dbinom(0 : 9, 14, 0.80))
## [1] 0.1298396

(a) Which of the Rejection Regions should we prefer? RR8 or RR10? Why?

Answer: we would prefer RR10 since the probability of a Type I error is considerably lower.

(b) In general, a hypothesis test can result in 2 different types of errors. Describe those 2 errors in this case where we are attempting to identify someone with a high degree of taste-discrimination ability. Which is more serious?

Answer: in the case of the triangle-taste test, a Type I error occurs when we reject the null hypothesis when it is true; that is, when we conclude that someone with no taste-discrimination ability actually has it. A Type II error occurs when we do not reject the null hypothesis when it is false; in other words,
when we conclude that someone who has ability does not actually have it after all. The consequences of committing a Type I error are more serious: the person hired as taster is less likely to discern problems with the product. The consequences of committing a Type II error are less serious since they mean that we fail to identify a subject who has taste-discrimination ability and we must continue interviewing and testing until the next qualified person applies for the position. Whereas the brewery can withstand a Type II error, it might not survive a Type I error.

(c) We saw in Chapter 9 that α and β are usually  in a trade-off relationship. That is, if we select one of two possible Rejection Regions—such as either RR8 or RR10 we can reduce α only if we are willing to have a higher β . Can you think of anything we might do to reduce both α and β simultaneously? What would that be?

Answer: it is possible to reduce both α and β simultaneously by increasing the sample size n. In this case, this would involve increasing the number of trials.