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

The 20 questions

1. What does the null hypothesis $H_0$ state in a chi-squared goodness of fit test?

   • Data fits distribution
   • Data is random
   • Variables are independent
   • Means are equal

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. In a chi-squared goodness of fit test, cells are pooled if the expected frequency is less than what value?

   • 0.05
   • 1
   • 5
   • 10

4. In a goodness of fit test with $k$ cells and $p$ estimated parameters, what are the degrees of freedom?

   • $k - 1 - p$
   • $k - 1$
   • $k - p$
   • $k - 1 + p$

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. For $\chi^2 = 10.2$ with $4$ degrees of freedom (critical value $9.488$), what is the decision at $5\%$ level?

    • Do not reject $H_0$
    • Reject $H_1$
    • Accept $H_0$
    • Reject $H_0$

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 must expected frequencies be at least $5$ in a chi-squared goodness of fit test?

    • Valid approximation
    • Ensure equal probabilities
    • Avoid zero division
    • Increase sample mean

16. A hypothesis test yields a p-value of 0.03. At which significance levels is H0 rejected?

    • Both 1% and 5%
    • 1% but not 5%
    • Neither 1% nor 5%
    • 5% but not 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 adjacent cells merged in a chi-squared goodness of fit test?

    • Reduce sample size
    • Increase observed frequencies
    • Increase expected frequencies
    • Increase variance

20. A chi-squared test statistic is 7.50 with 3 degrees of freedom. What is the decision at 5% significance?

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

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