SC1-SC2 Type I and Type II errors and the power of a test: AQA Further Maths, Unit 4

The 20 questions

1. A Type I error in a hypothesis test is:

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

2. A Type II error in a hypothesis test is:

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

3. The power of a test is defined as:

   • 1 - P(Type II error)
   • P(Type II error)
   • P(Type I error)
   • 1 - P(Type I error)

4. The probability of a Type I error for a test is equal to:

   • The probability of accepting H0 when H1 is true
   • The probability of accepting H0 under H1
   • The power of the test
   • The probability of the critical region under H0

5. Increasing the significance level of a test increases the probability of which error?

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

6. In a medical context, what would a Type I error mean for a test of a new drug?

   • Failing to detect a real effect
   • Concluding the drug has no effect when it does
   • Measuring the wrong dose
   • Concluding the drug works when it does not

7. Which error is controlled directly by the chosen significance level?

   • Power
   • Type I error
   • Type II error
   • The sample size

8. H0: lambda = 2 is tested against H1: lambda < 2 with critical region X = 0 for X ~ Po(lambda). What is the probability of a Type I error, to 4 decimal places?

   • 0.8647
   • 0.3679
   • 0.1353
   • 0.0183

9. For the same test, H0: lambda = 2 against H1: lambda < 2 with critical region X = 0. What is the power when lambda = 1, to 4 decimal places?

   • 0.6321
   • 0.1353
   • 0.3679
   • 0.6065

10. For the same test with critical region X = 0 and true lambda = 1, what is the probability of a Type II error, to 4 decimal places?

    • 0.1353
    • 0.6065
    • 0.3679
    • 0.6321

11. H0: lambda = 5 against H1: lambda > 5 with critical region X >= 10. What is the probability of a Type I error, to 4 decimal places?

    • 0.0318
    • 0.05
    • 0.0681
    • 0.9682

12. A one-tailed z-test is carried out at the 5% significance level, using a normal distribution with known variance. What is the probability of a Type I error?

    • 0.01
    • 0.025
    • 0.05
    • 0.95

13. A two-tailed z-test is carried out at the 1% significance level. What is the probability of a Type I error?

    • 0.5
    • 0.99
    • 0.005
    • 0.01

14. A test has P(Type II error) = 0.3 when the alternative is true. What is its power?

    • 0.5
    • 0.3
    • 0.1
    • 0.7

15. A quality control test concludes a machine is faulty when it is actually working. Which error is this?

    • Type II error
    • No error
    • Type I error
    • Power error

16. A Poisson test has H0: lambda = 3 and reject H0 when X >= 6, with X ~ Po(3) under H0. What is the probability of a Type I error, to 4 decimal places?

    • 0.0161
    • 0.0508
    • 0.9161
    • 0.0839

17. For the test of H0: lambda = 2 against H1: lambda < 2 with critical region X = 0, what is the power when lambda = 0.5, to 4 decimal places?

    • 0.3679
    • 0.8647
    • 0.1353
    • 0.6065

18. For the same test when lambda = 0.5, what is the probability of a Type II error, to 4 decimal places?

    • 0.6321
    • 0.3935
    • 0.3679
    • 0.6065

19. If the significance level of a test is reduced from 0.05 to 0.01 with the alternative hypothesis fixed, what happens to the probability of a Type II error?

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

20. For the test of H0: lambda = 4 against H1: lambda < 4 with critical region X <= 1, what is the probability of a Type I error, to 4 decimal places?

    • 0.5000
    • 0.0183
    • 0.0916
    • 0.9084

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