| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Explain Type I or II error |
| Difficulty | Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with standard follow-up questions about Type I and Type II errors. Part (i) requires routine application of the binomial test procedure (stating hypotheses, calculating P(X≥3) under H₀, comparing to 10% significance level). Parts (ii) and (iii) test standard definitions that are commonly examined. The small sample size (n=10) makes calculations manageable, and all steps follow a well-practiced template with no novel problem-solving required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: P(6) = \frac{1}{6}\) \(\quad\) \(H_1: P(6) > \frac{1}{6}\) | B1 | Allow "p" |
| \(1 - ((\frac{5}{6})^{10} + 10(\frac{1}{6})(\frac{5}{6})^9 + {}^{10}C_2(\frac{1}{6})^2(\frac{5}{6})^8)\) | M1 | Allow 1 term omitted or extra or incorrect |
| \(= 0.225\) (3 sfs) | A1 | |
| \(0.225 > 0.1\) | M1 | Allow correct comparison with 0.9, and recovery of previous then M1A1 possible |
| No evidence that die biased | A1ft [5] | Allow "Accept die not biased." In context. SR Calc just \(P(3)\) max score B1M0A0M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(4 \text{ or more sixes})\) | M1 | Idea of \(1 - \Sigma\) of terms oe compared with 0.1 |
| \(= 1 - ((\frac{5}{6})^{10} + 10(\frac{1}{6})(\frac{5}{6})^9 + {}^{10}C_2(\frac{1}{6})^2(\frac{5}{6})^8 + {}^{10}C_3(\frac{1}{6})^3(\frac{5}{6})^7)\) | M1 | \(1 - \Sigma\) of appropriate no. terms oe compared with 0.1 |
| \(= 0.0697\) or \(0.0698\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Concluding die is fair when die is biased | B1 [1] | Must be in context |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: P(6) = \frac{1}{6}$ $\quad$ $H_1: P(6) > \frac{1}{6}$ | B1 | Allow "p" |
| $1 - ((\frac{5}{6})^{10} + 10(\frac{1}{6})(\frac{5}{6})^9 + {}^{10}C_2(\frac{1}{6})^2(\frac{5}{6})^8)$ | M1 | Allow 1 term omitted or extra or incorrect |
| $= 0.225$ (3 sfs) | A1 | |
| $0.225 > 0.1$ | M1 | Allow correct comparison with 0.9, and recovery of previous then M1A1 possible |
| No evidence that die biased | A1ft [5] | Allow "Accept die not biased." In context. SR Calc just $P(3)$ max score B1M0A0M1A0 |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(4 \text{ or more sixes})$ | M1 | Idea of $1 - \Sigma$ of terms oe compared with 0.1 |
| $= 1 - ((\frac{5}{6})^{10} + 10(\frac{1}{6})(\frac{5}{6})^9 + {}^{10}C_2(\frac{1}{6})^2(\frac{5}{6})^8 + {}^{10}C_3(\frac{1}{6})^3(\frac{5}{6})^7)$ | M1 | $1 - \Sigma$ of appropriate no. terms oe compared with 0.1 |
| $= 0.0697$ or $0.0698$ | A1 [3] | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Concluding die is fair when die is biased | B1 [1] | Must be in context |
---6 It is claimed that a certain 6-sided die is biased so that it is more likely to show a six than if it was fair. In order to test this claim at the $10 \%$ significance level, the die is thrown 10 times and the number of sixes is noted.\\
(i) Given that the die shows a six on 3 of the 10 throws, carry out the test.
On another occasion the same test is carried out again.\\
(ii) Find the probability of a Type I error.\\
(iii) Explain what is meant by a Type II error in this context.
\hfill \mbox{\textit{CAIE S2 2010 Q6 [9]}}