| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find sample size for test |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard binomial test setup, a single probability calculation to verify significance, and finding a sample size by trial (or solving an inequality). All techniques are routine for S2 level with no novel insight required, making it slightly easier than average. |
| Spec | 2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: P(6) = \frac{1}{6}\) \(\quad\) \(H_1: P(6) < \frac{1}{6}\) | B1 [1] | Allow \(H_0: p = \frac{1}{6}\) \(\quad\) \(H_1: p < \frac{1}{6}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(\frac{5}{6}\right)^{15}\) | M1 | |
| \(= 0.065 > 0.05\) | A1 [2] | Correct result and comparison needed for A1. SR if 2 tail test followed allow A1 for \(0.065 > 0.025\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(\frac{5}{6}\right)^{16} = 0.054\) and \(\left(\frac{5}{6}\right)^{17} = 0.045\) | M1 | both |
| Smallest \(n\) is 17 | A1 [2] | No errors seen |
| OR | ||
| \(\left(\frac{5}{6}\right)^n < 0.05\) and attempt to solve | M1 | |
| \(n\ln\left(\frac{5}{6}\right) < \ln 0.05\) | ||
| smallest \(n\) is 17 | A1 |
## Question 2:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: P(6) = \frac{1}{6}$ $\quad$ $H_1: P(6) < \frac{1}{6}$ | B1 [1] | Allow $H_0: p = \frac{1}{6}$ $\quad$ $H_1: p < \frac{1}{6}$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(\frac{5}{6}\right)^{15}$ | M1 | |
| $= 0.065 > 0.05$ | A1 [2] | Correct result and comparison needed for A1. SR if 2 tail test followed allow A1 for $0.065 > 0.025$ |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(\frac{5}{6}\right)^{16} = 0.054$ and $\left(\frac{5}{6}\right)^{17} = 0.045$ | M1 | both |
| Smallest $n$ is 17 | A1 [2] | No errors seen |
| **OR** | | |
| $\left(\frac{5}{6}\right)^n < 0.05$ and attempt to solve | M1 | |
| $n\ln\left(\frac{5}{6}\right) < \ln 0.05$ | | |
| smallest $n$ is 17 | A1 | |
---2 A die has six faces numbered $1,2,3,4,5,6$. Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.\\
(i) State suitable null and alternative hypotheses for Manjit's test.\\
(ii) There are no sixes in the first 15 throws. Show that this result is not significant at the $5 \%$ level.\\
(iii) Find the smallest value of $n$ such that, if there are no sixes in the first $n$ throws, this result is significant at the 5\% level.
\hfill \mbox{\textit{CAIE S2 2016 Q2 [5]}}