Lesson SB5

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

20 questions

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Lesson SB5, Hypothesis test for a Poisson mean: 20 multiple choice questions for the AQA Further Maths (7367), Unit 4: Optional application 2: statistics, written with Revision Ninja.

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The 20 questions

  1. In a hypothesis test about a Poisson mean from a single observation, what is the null hypothesis typically?

    • The parameter has the stated value, such as lambda = 4
    • The variance is unknown
    • The observation is always zero
    • The alternative hypothesis is always true

  2. What is the significance level of a test?

    • The probability of accepting a false null hypothesis
    • The sample mean
    • The probability of rejecting the null hypothesis when it is true
    • The power of the test

  3. When a one-tailed test is carried out for a Poisson mean, what determines the critical region?

    • The tail indicated by the alternative hypothesis
    • The centre of the distribution
    • The value of the observation alone
    • Both tails, always equally

  4. For a single observation X from Po(lambda) under H0, what quantity is compared with the significance level to decide whether to reject H0?

    • The probability of a result at least as extreme as X, under H0
    • The variance of X only
    • The mean of X only
    • The median of X only

  5. A single observation X from Po(lambda) is used to test H0: lambda = 4 against H1: lambda < 4. P(X <= 1) for lambda = 4 is 0.0916 to 4 decimal places. At the 5% level, what is the conclusion if X = 1?

    • Accept H1 with certainty
    • Reject H0 since 0.0916 is less than 0.05
    • Reject H0 since X = 1 is small
    • Do not reject H0 since 0.0916 is greater than 0.05

  6. A test of H0: lambda = 4 against H1: lambda < 4 at the 5% level observes X = 0. Taking P(X = 0) = e^(-4) = 0.0183, is H0 rejected?

    • Yes, since 0.0183 is less than 0.05
    • Yes, since 0.0183 is greater than 0.05
    • No, since 0.0183 is greater than 0.05
    • No, since X = 0 is not in the critical region

  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. A Poisson test of H0: lambda = 5 against H1: lambda > 5 observes X = 10. Using P(X >= 10) = 0.0318, what is the conclusion at the 5% level?

    • Accept H0 as the mean is confirmed
    • Reject H1 in favour of H0
    • Reject H0 in favour of H1
    • No conclusion can be drawn from one observation

  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 statement best describes a Type I error in a Poisson test?

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

  11. X ~ Po(lambda) under H0: lambda = 2 is tested against H1: lambda > 2 at the 5% level. For X = 6, P(X >= 6) = 0.0166 to 4 decimal places. What is the conclusion?

    • Reject H0 since 0.0166 is greater than 0.05
    • Reject H0 since 0.0166 is less than 0.05
    • Accept H1 since the mean is 6
    • Do not reject H0 since 6 is not large

  12. A Poisson test uses H0: lambda = 3 against H1: lambda not equal to 3 (two-tailed). How does the method differ from a one-tailed test at the same significance level?

    • Each tail is tested at the full significance level
    • The critical region is always the upper tail only
    • Each tail is tested at half the significance level
    • The test uses a normal approximation with variance 3

  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. A Poisson test has H0: lambda = 6 and H1: lambda < 6. P(X <= 2) = 0.0620 to 4 decimal places. Is X = 2 significant at the 5% level?

    • No, since 0.0620 is greater than 0.05
    • Yes, since 0.0620 is greater than 0.05
    • No, since 0.0620 is less than 0.05
    • Yes, since 0.0620 is less than 0.05

  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. A Poisson test of H0: lambda = 2 against H1: lambda < 2 uses observation X = 0. P(X = 0) = e^(-2) = 0.1353. Is H0 rejected at the 5% level?

    • Yes, since 0.1353 is greater than 0.05
    • No, since 0.1353 is less than 0.05
    • No, since 0.1353 is greater than 0.05
    • Yes, since 0.1353 is less than 0.05

  17. A Poisson test of H0: lambda = 3 against H1: lambda > 3 at the 5% level. For X = 7, P(X >= 7) = 0.0335 to 4 decimal places. What is the result?

    • Accept H0 since 7 is near the mean
    • No conclusion can be drawn
    • Reject H0 in favour of H1
    • Accept H1 with certainty

  18. A Poisson test of H0: lambda = 4 against H1: lambda > 4 at the 5% level observes X = 8. P(X >= 8) = 0.0511 to 4 decimal places. What is the conclusion?

    • Accept H1, since X = 8 exceeds the mean
    • Reject H0, since 0.0511 is less than 0.05
    • Do not reject H0, since 0.0511 is greater than 0.05
    • Reject H0, since 8 is greater than 4

  19. A Poisson test of H0: lambda = 10 against H1: lambda < 10 observes X = 3, with P(X <= 3) = 0.0103 to 4 decimal places. At the 5% level, is the result significant?

    • Yes, since 0.0103 is less than 0.05
    • Yes, since 0.0103 is greater than 0.05
    • No, since 0.0103 is less than 0.05 only at the 1% level
    • No, since 0.0103 is greater than 0.05

  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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