3B.8.1 Type I and Type II errors Quiz: Pearson Edexcel Further Maths, Unit 17

The 20 questions

1. What is a Type I error in a hypothesis test?

   • Rejecting H0 when H0 is true
   • Rejecting H1 when H1 is true
   • Accepting H0 when H0 is true
   • Accepting H0 when H0 is false

2. What is a Type II error in a hypothesis test?

   • Rejecting H0 when H0 is true
   • Rejecting H1 when H1 is true
   • Rejecting H0 when H1 is true
   • Accepting H0 when H1 is true

3. What is the power of a test?

   • 1 minus P(Type II error), which is P(reject H0 | H1 is true)
   • P(accept H0 | H0 is true), which is the probability of a correct decision
   • The significance level, which is the largest probability of a Type II error
   • P(Type I error), the probability of rejecting H0 when H0 is actually true

4. What does the significance level of a test equal?

   • The power of the test
   • 1 minus P(Type I error)
   • P(Type II error)
   • P(Type I error)

5. What does the power function of a test give?

   • The probability of H0 as a function of the sample size only
   • The mean of the test statistic
   • The probability of rejecting H0 as a function of the parameter
   • The p-value for every sample

6. X ~ Po(lambda). H0: lambda = 2, reject H0 if X >= 5. What is P(Type I error)?

   • About 0.947
   • About 0.017
   • About 0.135
   • About 0.053

7. X ~ B(10, 0.5). H0: p = 0.5, H1: p > 0.5, reject if X >= 9. What is P(Type I error)?

   • About 0.011
   • About 0.5
   • About 0.001
   • About 0.055

8. For the test in which X ~ B(10, 0.5) and H0 is rejected if X >= 9, what is the power when p = 0.7?

   • About 0.5
   • About 0.851
   • About 0.011
   • About 0.149

9. For the same test with p = 0.7 as the true value, what is the probability of a Type II error?

   • About 0.5
   • About 0.989
   • About 0.851
   • About 0.149

10. X ~ Po(lambda). H0: lambda = 1, reject H0 if X >= 3. What is P(Type I error)?

    • About 0.080
    • About 0.027
    • About 0.920
    • About 0.184

11. For the test in which H0 is lambda = 1 and H0 is rejected if X >= 3, what is the power when lambda = 3?

    • About 0.080
    • About 0.250
    • About 0.423
    • About 0.577

12. Which quantity does the probability of a Type I error depend on?

    • Only H0 and the critical region
    • The value of the alternative parameter
    • The sample size, but only if H1 is true
    • The power of the test

13. For a fixed significance level, what generally happens to power when the sample size n increases?

    • The significance level is reduced to zero
    • Power is unchanged
    • Power increases
    • Power decreases

14. In a court trial, convicting an innocent defendant corresponds to which error?

    • A Type II error
    • A power error
    • A Type I error
    • No error

15. X ~ B(8, 0.5). H0: p = 0.5, H1: p < 0.5, reject if X <= 1. What is P(Type I error)?

    • About 0.965
    • About 0.004
    • About 0.035
    • About 0.145

16. For the test in which X ~ B(8, 0.5) and H0 is rejected if X <= 1, what is the probability of a Type II error when p = 0.3?

    • About 0.745
    • About 0.965
    • About 0.255
    • About 0.035

17. A test has P(Type I error) = 0.05 and P(Type II error) = 0.2 for a given alternative. What is the power against that alternative?

    • 0.05
    • 0.2
    • 0.8
    • 0.95

18. Which error is controlled directly by the significance level?

    • Both errors equally
    • Neither error
    • Type II error
    • Type I error

19. If the critical region is moved outward to reduce the Type I error, what happens to the Type II error?

    • It is unchanged
    • It decreases
    • It becomes zero
    • It increases

20. When H1 is true and the true value is far from the value under H0, how does the power of the test generally behave?

    • It is smaller
    • It is zero
    • It equals the significance level
    • It is larger

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