Lesson 3B.6.1

3B.6.1 Goodness of fit tests for discrete distributions Quiz: Pearson Edexcel Further Maths, Unit 15

20 questions

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Lesson 3B.6.1, Goodness of fit tests for discrete distributions: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 15: Goodness of fit, written with Revision Ninja.

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The 20 questions

  1. In a goodness of fit test, what is the null hypothesis?

    • The data follow the stated distribution
    • The data follow a different distribution from the one stated
    • The sample size is large
    • The population mean is zero

  2. What is the test statistic used in a goodness of fit test?

    • Sum of (O_i - E_i)^2 / E_i
    • Sum of (O_i - E_i)^2
    • Sum of (O_i - E_i)/E_i
    • Sum of O_i^2 / E_i

  3. When should neighbouring cells be combined in a chi-squared goodness of fit test?

    • Never
    • When the observed frequency in a cell is greater than 5
    • When the expected frequency in a cell is less than 5
    • When the expected frequency exceeds 10

  4. What is the rule for degrees of freedom when parameters are estimated from the data?

    • Number of cells minus 1 minus number of estimated parameters
    • Number of observations minus 1, regardless of how many cells the table has
    • Number of cells plus the number of parameters estimated from the data
    • Number of cells minus 1, ignoring any parameters that were estimated from the data

  5. A fair die is rolled 60 times and the counts are compared with the expected counts of 10 per face. What are the degrees of freedom?

    • 6
    • 5
    • 60
    • 4

  6. A Poisson model has lambda estimated from the data, and 6 cells remain after combining. What are the degrees of freedom?

    • 5
    • 3
    • 6
    • 4

  7. An observed count is 18 and the expected count is 12. What is its contribution to the chi-squared statistic?

    • 1.5
    • 18
    • 3
    • 6

  8. Observed frequencies are 20, 30 and 50, with expected frequencies 25, 25 and 50. What is the chi-squared statistic?

    • 4
    • 1
    • 0
    • 2

  9. A die is rolled 120 times with observed counts 25, 18, 20, 17, 22, 18 and expected count 20 per face. What is the chi-squared statistic to 1 decimal place?

    • 5.0
    • 2.3
    • 3.5
    • 0.5

  10. What is the 5% critical value of the chi-squared distribution with 4 degrees of freedom, to 3 decimal places?

    • 7.815
    • 11.070
    • 5.991
    • 9.488

  11. A goodness of fit test gives chi-squared = 10.2 with 4 degrees of freedom. What is the decision at the 5% level?

    • Do not reject H0, since 10.2 < 11.070
    • Do not reject H0, since 10.2 > 9.488
    • Reject H0, since 10.2 < 11.070
    • Reject H0, since 10.2 > 9.488

  12. A Poisson distribution with lambda = 1 is fitted to n = 100 observations. What is the expected count for X = 0, to 1 decimal place?

    • 1.0
    • 100.0
    • 36.8
    • 63.2

  13. A binomial B(4, 0.5) model is fitted to 200 observations. What is the expected count for X = 2?

    • 75
    • 37.5
    • 50
    • 100

  14. A goodness of fit test has 7 cells and no parameters estimated. What are the degrees of freedom?

    • 5
    • 8
    • 7
    • 6

  15. Why does the chi-squared approximation require expected frequencies of at least 5?

    • So that the test statistic is approximately chi-squared distributed
    • So that the data are approximately normal and the mean is a good estimate
    • So that the observed frequencies are integers and can be compared directly
    • So that the p-value equals exactly 0.05 whatever the observed data are

  16. A test gives a p-value of 0.03. What is the conclusion?

    • Reject H0 at both 5% and 1%
    • Do not reject H0 at 5% but reject at 1%
    • Do not reject H0 at either level
    • Reject H0 at 5% but not at 1%

  17. A binomial model with n known has p estimated from the data. The table has k cells. What are the degrees of freedom?

    • k - 1
    • k - 3
    • k - 2
    • k

  18. What distribution does the goodness of fit test statistic approximately follow under H0?

    • Poisson
    • Normal
    • Chi-squared
    • Binomial

  19. Why are cells combined in a goodness of fit test?

    • To reduce the degrees of freedom to zero so the test becomes trivial
    • To increase the observed frequencies so that they match the expected values
    • To keep expected frequencies large enough for the approximation
    • To make the data normal so that a normal test can be used instead

  20. A chi-squared statistic is 7.5 with 3 degrees of freedom. What is the decision at the 5% level?

    • Do not reject H0, since 7.5 < 7.815
    • Do not reject H0, since 7.5 > 7.815
    • Reject H0, since 7.5 > 7.815
    • Reject H0, since 7.5 < 7.815

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