# Statistics with R

## Student Resources

# 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: RR_{7 }= {7, 8, 9, 10, 11} and RR_{9 }= {9, 10, 11}. Assuming that the rejection region is RR_{7 }= {7, 8, 9, 10, 11}, what is the probability of Type I error, α?

- 0.03862894 X
- 0.00743508
- 0.03725719
- 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 RR_{7 }= {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?

- 0.0073307
- 0.2503683
- 0.2889973 X
- 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 RR_{9 }= {9, 10, 11}, what is the probability of Type I error, α?

- 0.002567901
- 0.001975309
- 0.001646091
- 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 RR_{9 }= {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?

- 0.7658893 X
- 0.5744170
- 0.8424783
- 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?

- 0.002057613
- 0.001371742 X
- 0.003429355
- 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: RR_{10 }= {10, 11, 12, 13, 14, 15, 16} and RR_{12 }= {12, 13, 14, 15, 16}. With a rejection region of RR_{10 }= {10, 11, 12, 13, 14, 15, 16}, what is the probability of Type I error, α?

- 0.00403954
- 0.04996248
- 0.12650070
- 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 RR_{10 }= {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?

- 0.17531340
- 0.07955725 X
- 0.15940540
- 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 RR_{12 }= {12, 13, 14, 15, 16}, what is the probability of Type I error, α?

- 0.0007924645 X
- 0.0040395410
- 0.0001159903
- 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 RR_{12 }= {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?

- 0.1896546
- 0.5500959
- 0.3698138 X
- 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?

- 0.049962480
- 0.015945490
- 0.000792464
- 0.004039541 X

**Solution:**

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

[1] 0.004039541

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

[1] 0.004039541