| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Interpret test conclusion or error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring understanding of Type I error definition (rejecting H₀ when true) and recognizing which error type is possible given observed data. Part (i) is a direct binomial probability calculation P(X < 17 | p = 0.9), and part (ii) tests conceptual understanding that Type II errors occur when we fail to reject H₀. Both parts are standard textbook applications with no novel problem-solving required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0:\) Rate \(= 0.9\); \(H_1:\) Rate \(< 0.9\) | B1 | \(p = 0.9\); \(p < 0.9\); Use of \(B(20, 0.1)\) |
| \(1 - P(17, 18, 19, 20)\) | M1 | Allow \(1-P(18,19,20)\) or \(1-P(16,17,18,19,20)\) |
| \(1-(^{20}C_{17} \times 0.1^3 \times 0.9^{17} + ^{20}C_{18} \times 0.1^2 \times 0.9^{18} + 20 \times 0.1 \times 0.9^{19} + 0.9^{20})\) | M1 | |
| \(= 0.133\) (3 sf) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Type II | B1 | or Stephan will conclude standard not fallen |
| \(H_0\) will not be rejected | B1 [2] | No contradictions |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0:$ Rate $= 0.9$; $H_1:$ Rate $< 0.9$ | B1 | $p = 0.9$; $p < 0.9$; Use of $B(20, 0.1)$ |
| $1 - P(17, 18, 19, 20)$ | M1 | Allow $1-P(18,19,20)$ or $1-P(16,17,18,19,20)$ |
| $1-(^{20}C_{17} \times 0.1^3 \times 0.9^{17} + ^{20}C_{18} \times 0.1^2 \times 0.9^{18} + 20 \times 0.1 \times 0.9^{19} + 0.9^{20})$ | M1 | |
| $= 0.133$ (3 sf) | A1 [4] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Type II | B1 | or Stephan will conclude standard not fallen |
| $H_0$ will not be rejected | B1 [2] | No contradictions |
---6 Stephan is an athlete who competes in the high jump. In the past, Stephan has succeeded in $90 \%$ of jumps at a certain height. He suspects that his standard has recently fallen and he decides to carry out a hypothesis test to find out whether he is right. If he succeeds in fewer than 17 of his next 20 jumps at this height, he will conclude that his standard has fallen.\\
(i) Find the probability of a Type I error.\\
(ii) In fact Stephan succeeds in 18 of his next 20 jumps. Which of the errors, Type I or Type II, is possible? Explain your answer.
\hfill \mbox{\textit{CAIE S2 2014 Q6 [6]}}