| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find or state significance level |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial hypothesis testing with standard procedures. Part (i) requires calculating P(X ≥ 10) under H₀: p=0.5, which is direct computation. Part (ii) involves finding the critical region using normal approximation or tables at a standard significance level. Both parts are routine textbook exercises requiring no novel insight, though slightly easier than average due to the mechanical nature of the calculations. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(> 9 \text{ Heads} \mid \text{unbiased}) = {}^{12}C_{10} \times 0.5^{10} \times 0.5^2 + 12 \times 0.5^{11} \times 0.5 + 0.5^{12}\) | M1 | Allow Bin \(P(X = 9,10,11,12)\) correct or \(1 - P(X = (9),10,11,12)\) any \(p/q\) |
| M1 | Allow Bin \(P(X = 9,10,11,12)\) correct \(p/q\) | |
| \(= 0.0193\); Level is \(1.93\%\) or \(1.9\%\) | A1 (3) | Allow \(2\%\) if correct working seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B(100, 0.5) \approx N(50, 25)\) | B1 | Or proportion method \(N(0.5, 0.0025)\) |
| \(\frac{x - 0.5 - 50}{\sqrt{25}} = z\) | M1 | Allow with wrong or no cc or no \(\sqrt{}\) (cc for proportion method \(0.5/100\)) |
| \(z = 1.645\) | B1 | + only (consistent with their standardisation) |
| \(x = 58.7\) | A1 | |
| Rejection region is \(> 59\) | A1ft (5) | or \(> 58\) (region and integer required) |
## Question 5:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(> 9 \text{ Heads} \mid \text{unbiased}) = {}^{12}C_{10} \times 0.5^{10} \times 0.5^2 + 12 \times 0.5^{11} \times 0.5 + 0.5^{12}$ | M1 | Allow Bin $P(X = 9,10,11,12)$ correct or $1 - P(X = (9),10,11,12)$ any $p/q$ |
| | M1 | Allow Bin $P(X = 9,10,11,12)$ correct $p/q$ |
| $= 0.0193$; Level is $1.93\%$ or $1.9\%$ | A1 (3) | Allow $2\%$ if correct working seen |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B(100, 0.5) \approx N(50, 25)$ | B1 | Or proportion method $N(0.5, 0.0025)$ |
| $\frac{x - 0.5 - 50}{\sqrt{25}} = z$ | M1 | Allow with wrong or no cc or no $\sqrt{}$ (cc for proportion method $0.5/100$) |
| $z = 1.645$ | B1 | + only (consistent with their standardisation) |
| $x = 58.7$ | A1 | |
| Rejection region is $> 59$ | A1ft (5) | or $> 58$ (region and integer required) |
---5 (i) Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.\\
(ii) Deng throws another coin 100 times in order to test, at the $5 \%$ significance level, whether it is biased towards Heads. Find the rejection region for this test.
\hfill \mbox{\textit{CAIE S2 2012 Q5 [8]}}