SB5 Hypothesis test for a Poisson mean Quiz: AQA Further Maths, Unit 4

The 20 questions

1. What form does the null hypothesis take in a hypothesis test for a Poisson mean?

   • lambda = c
   • lambda != c
   • lambda < c
   • lambda > c

2. What is the definition of the significance level of a hypothesis test?

   • Power of test
   • P(H0 is true)
   • P(Type I error)
   • P(Type II error)

3. In a one-tailed Poisson hypothesis test, what determines which tail forms the critical region?

   • Alternative hypothesis
   • Null hypothesis
   • Significance level
   • Sample mean

4. In a Poisson hypothesis test, what probability value is compared directly against the significance level?

   • p-value
   • Poisson mean
   • Test statistic
   • Critical value

5. For H0: λ = 4 against H1: λ < 4 at 5% significance, P(X ≤ 1) = 0.0916. What is the outcome?

   • Invalid test
   • Accept H1
   • Reject H0
   • Do not reject H0

6. For H0: λ = 4 against H1: λ < 4 at 5% significance, P(X = 0) = 0.0183. Is H0 rejected?

   • Yes
   • Inconclusive
   • No
   • Only at 1%

7. A single observation X from Po(5) is used to test H0: lambda = 5 against H1: lambda > 5. P(X >= 9) = 0.0681 and P(X >= 10) = 0.0318 to 4 decimal places. At the 5% level, what is the critical region?

   • X >= 10
   • X >= 11
   • X >= 9
   • X >= 8

8. For H0: λ = 5 against H1: λ > 5 at 5% significance, P(X ≥ 10) = 0.0318. What is the decision?

   • Do not reject H0
   • Accept H0
   • Reject H0
   • Inconclusive

9. Under H0: lambda = 5 for X ~ Po(5), what is P(X <= 0) to 4 decimal places?

   • 0.0404
   • 0.1755
   • 0.0337
   • 0.0067

10. Which of the following defines a Type I error in a hypothesis test?

    • Accepting true H0
    • Rejecting true H0
    • Accepting false H0
    • Rejecting false H0

11. For H0: λ = 2 against H1: λ > 2 at 5% significance, P(X ≥ 6) = 0.0166. What is the outcome?

    • Accept H0
    • Reject H0
    • Inconclusive
    • Do not reject H0

12. In a two-tailed Poisson test at significance level α, what is the tail probability?

    • α
    • 2α
    • α / 2
    • 1 − α

13. A test of H0: lambda = 4 against H1: lambda < 4 at the 5% level has critical region X <= 0. What is the actual significance level?

    • 5%
    • 0.67%
    • 9.16%
    • 1.83%

14. Testing H0: λ = 6 against H1: λ < 6 at 5% significance gives P(X ≤ 2) = 0.0620. Is it significant?

    • No
    • Yes
    • Only at 1%
    • Inconclusive

15. A test of H0: lambda = 5 against H1: lambda > 5 at the 5% level has critical region X >= 10. What is the probability of a Type I error?

    • 0.0318
    • 0.9682
    • 0.05
    • 0.0681

16. Testing H0: λ = 2 against H1: λ < 2 at 5% significance gives P(X = 0) = 0.1353. Is H0 rejected?

    • Yes
    • Reject H1
    • No
    • Inconclusive

17. For H0: λ = 3 against H1: λ > 3 at 5% significance, P(X ≥ 7) = 0.0335. What is the result?

    • Do not reject H0
    • Inconclusive
    • Reject H0
    • Accept H0

18. For H0: λ = 4 against H1: λ > 4 at 5% significance, P(X ≥ 8) = 0.0511. What is the outcome?

    • Accept H1
    • Reject H0
    • Do not reject H0
    • Inconclusive

19. Testing H0: λ = 10 against H1: λ < 10 at 5% significance gives P(X ≤ 3) = 0.0103. Is it significant?

    • Yes
    • Only at 1%
    • Inconclusive
    • No

20. A two-tailed Poisson test of H0: lambda = 5 against H1: lambda not equal to 5 is carried out at the 5% level, so each tail has probability at most 0.025. What is the upper critical region?

    • X >= 10
    • X >= 9
    • X >= 11
    • X >= 12

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