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

1.  A triangle taste test consists of 11 identical trials on which the subject attempts to identify the odd sample on each trial. Assume that there are 2 possible rejection regions: RR7 = {7, 8, 9, 10, 11} and RR9 = {9, 10, 11}. Assuming that the rejection region is RR7 = {7, 8, 9, 10, 11}, what is the probability of Type I error, α?

1. 0.03862894 X
2. 0.00743508
3. 0.03725719
4. 0.00194642

Solution:

> 1 - pbinom(6, 11, 1/3)

[1] 0.03862894

or

> sum(dbinom(7 : 11, 11, 1/3))

[1] 0.03862894

2.  With a rejection region of RR7 = {7, 8, 9, 10, 11}, what is the probability of a Type II error, β, if the subject has a probability of p=2/3 of identifying the odd sample?

1. 0.0073307
2. 0.2503683
3. 0.2889973 X
4. 0.2754052

Solution:

> pbinom(6, 11, 2/3)

[1] 0.2889973

or

> sum(dbinom(0 : 6, 11, 2/3))

[1] 0.2889973

3.  With a rejection region of RR9 = {9, 10, 11}, what is the probability of Type I error, α?

1. 0.002567901
2. 0.001975309
3. 0.001646091
4. 0.001371742 X

Solution:

> 1 - pbinom(8, 11, 1/3)

[1] 0.001371742

or

> sum(dbinom(9 : 11, 11, 1/3))

[1] 0.001371742

4.  With a rejection region of RR9 = {9, 10, 11}, what is the probability of a Type II error, β, if the subject has a probability of p=2/3 of identifying the odd sample?

1. 0.7658893 X
2. 0.5744170
3. 0.8424783
4. 0.7275949

Solution:

> pbinom(8, 11, 2/3)

[1] 0.7658893

or

> sum(dbinom(0 : 8, 11, 2/3))

[1] 0.7658893

5.  What is the p-value for a subject who achieves x = 9 correct identifications in n=11 trials?

1. 0.002057613
2. 0.001371742 X
3. 0.003429355
4. 0.001097394

Solution:

> 1 - pbinom(8, 11, 1/3)

[1] 0.001371742

or

> sum(dbinom(9 : 11, 11, 1/3))

[1] 0.001371742

6.   Another triangle taste test consists of 16 identical trials on which the subject attempts to identify the odd sample on each trial. Assume that there are 2 possible rejection regions: RR10 = {10, 11, 12, 13, 14, 15, 16} and RR12 = {12, 13, 14, 15, 16}. With a rejection region of RR10 = {10, 11, 12, 13, 14, 15, 16}, what is the probability of Type I error, α?

1. 0.00403954
2. 0.04996248
3. 0.12650070
4. 0.01594549 X

Solution:

> 1 - pbinom(9, 16, 1/3)

[1] 0.01594549

or

> sum(dbinom(10 : 16, 16, 1/3))

[1] 0.01594549

7.  With a rejection region of RR10 = {10, 11, 12, 13, 14, 15, 16}, what is the probability of a Type II error, β, if the subject has a probability of p=0.75 of identifying the odd sample?

1. 0.17531340
2. 0.07955725 X
3. 0.15940540
4. 0.02665733

Solution:

> pbinom(9, 16, 0.75)

[1] 0.07955725

or

> sum(dbinom(0 : 9, 16, 0.75))

[1] 0.07955725

8.  With a rejection region of RR12 = {12, 13, 14, 15, 16}, what is the probability of Type I error, α?

1. 0.0007924645 X
2. 0.0040395410
3. 0.0001159903
4. 0.0159454900

Solution:

> 1 - pbinom(11, 16, 1/3)

[1] 0.0007924645

or

> sum(dbinom(12 : 16, 16, 1/3))

[1] 0.0007924645

9.  With a rejection region of RR12 = {12, 13, 14, 15, 16}, what is the probability of a Type II error, β, if the subject has a probability of p=0.75 of identifying the odd sample?

1. 0.1896546
2. 0.5500959
3. 0.3698138 X
4. 0.2017546

Solution:

> pbinom(11, 16, 0.75)

[1] 0.3698138

or

> sum(dbinom(0 : 11, 16, 0.75))

[1] 0.3698138

10.   What is the p-value for a subject who achieves x = 11 correct identifications in n=16 trials?

1. 0.049962480
2. 0.015945490
3. 0.000792464
4. 0.004039541 X

Solution:

> 1 - pbinom(10, 16, 1/3)

[1] 0.004039541

> sum(dbinom(11 : 16, 16, 1/3))

[1] 0.004039541