3B.4.2 Hypothesis tests for the binomial distribution Quiz: Pearson Edexcel Further Maths, Unit 13

The 20 questions

1. In a hypothesis test for a Binomial distribution, which parameter do the hypotheses state?

   • Number of trials n
   • Success probability p
   • Variance
   • Sample mean

2. Under H0 with success probability p0, what is the distribution of X in a binomial test with n trials?

   • Po(n p0)
   • B(n, p0)
   • N(n p0, 1)
   • Geo(p0)

3. X ~ B(10, 0.5). H0: p = 0.5, H1: p > 0.5. X = 8 is observed. What is P(X >= 8) and the decision at the 5% level?

   • 0.0107, reject H0
   • 0.1094, do not reject H0
   • 0.0547, do not reject H0
   • 0.0547, reject H0

4. X ~ B(10, 0.5). H0: p = 0.5, H1: p > 0.5. X = 9 is observed. What is P(X >= 9) and the decision at the 5% level?

   • 0.0547, reject H0
   • 0.0107, do not reject H0
   • 0.0107, reject H0
   • 0.0010, reject H0

5. For X ~ B(20, 0.3), testing H₁: p < 0.3 at 5% significance with X = 2, what is the decision?

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

6. For H1: p < p0 in a binomial test, which values of X form the critical region?

   • Values near n/2 only
   • Small values of X
   • All values of X
   • Large values of X

7. Testing H₁: p < 1/6 for 10 die rolls with 0 sixes observed at the 5% level, what is the decision?

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

8. Testing H₁: p > 0.25 for B(8, 0.25) at the 5% level with X = 4, what is the decision?

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

9. For $X \sim \text{B}(12, 0.5)$, $H_0: p=0.5$ against $H_1: p>0.5$. What is the decision at $5\%$ level when $x=10$?

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

10. X ~ B(20, 0.5). H0: p = 0.5, H1: p > 0.5, at the 5% level. What is the smallest value of X in the critical region?

    • 18
    • 15
    • 14
    • 16

11. For $X \sim \text{B}(10, 0.1)$, $H_0: p=0.1$ against $H_1: p>0.1$. What is the decision at $5\%$ level when $x=3$?

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

12. What is the maximum probability of making a Type I error for a hypothesis test at the $5\%$ significance level?

    • At most $0.05$
    • At most $0.95$
    • Exactly $0.05$
    • At least $0.05$

13. For $X \sim \text{B}(15, 0.2)$, $H_0: p=0.2$ against $H_1: p>0.2$. What is the decision at $5\%$ level when $x=6$?

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

14. For $X \sim \text{B}(20, 0.1)$, testing $H_0: p=0.1$ against $H_1: p<0.1$, what is the decision at $5\%$ level when $x=0$?

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

15. X ~ B(40, 0.25) under H0. What is the expected count E(X)?

    • 15
    • 10
    • 40
    • 0.25

16. Two-tailed test with H1: p is not equal to 0.5 and n = 10 at the 5% level. Which critical region is appropriate?

    • X = 5 only
    • X <= 1 or X >= 9
    • X <= 3 or X >= 7
    • X <= 2 or X >= 8

17. What decision is made in a hypothesis test if the observed test statistic lies in the critical region?

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

18. For $X \sim \text{B}(15, 0.5)$, $H_0: p=0.5$ against $H_1: p>0.5$. What is the decision at $5\%$ level when $x=12$?

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

19. For $X \sim \text{B}(20, 0.5)$, $H_0: p=0.5$ against $H_1: p<0.5$. What is the decision at $5\%$ level when $x=5$?

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

20. When H0 is true, what is the largest probability with which a discrete binomial test can wrongly reject H0 at the 5% level?

    • At least 0.5
    • Exactly 0.95
    • Exactly 0.5
    • At most 0.05

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