| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Moderate -0.3 Part (i) requires understanding that Type I error occurs when Hâ‚€ is true (p=1/6) and calculating P(X<2) using binomial distribution with n=25, which is straightforward application of tables or formula. Part (ii) is a standard confidence interval calculation using normal approximation. Both parts test routine statistical concepts with no novel problem-solving required, making this slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left(\dfrac{5}{6}\right)^{25} + 25\left(\dfrac{5}{6}\right)^{24}\left(\dfrac{1}{6}\right)\) | M1 | Allow end errors, but just \(P(2)\) implies M0; accept p/q mix |
| \(= 0.0629\) final answer | A1 | |
| Sig level \(= 6.29\%\) | B1ft [3] | ft their \(P(X \leq 1)\) with Binomial used; allow 6.3% or 6% |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(p) \approx \dfrac{0.09 \times 0.91}{100} = 0.000819\) | M1 | For \(pq/100\) seen (any \(p/q\)) (must be probs) |
| \(z = 1.96\) | B1 | |
| \(0.09 \pm z\sqrt{\dfrac{0.09 \times 0.91}{100}}\) | M1 | For correct form of CI (any \(p/q\)) (must be probs) |
| \(= 0.034\) to \(0.146\) (3 dps) | A1 [4] |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left(\dfrac{5}{6}\right)^{25} + 25\left(\dfrac{5}{6}\right)^{24}\left(\dfrac{1}{6}\right)$ | M1 | Allow end errors, but just $P(2)$ implies M0; accept p/q mix |
| $= 0.0629$ final answer | A1 | |
| Sig level $= 6.29\%$ | B1ft [3] | ft their $P(X \leq 1)$ with Binomial used; allow 6.3% or 6% |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(p) \approx \dfrac{0.09 \times 0.91}{100} = 0.000819$ | M1 | For $pq/100$ seen (any $p/q$) (must be probs) |
| $z = 1.96$ | B1 | |
| $0.09 \pm z\sqrt{\dfrac{0.09 \times 0.91}{100}}$ | M1 | For correct form of CI (any $p/q$) (must be probs) |
| $= 0.034$ to $0.146$ (3 dps) | A1 [4] | |
---3 Joshi suspects that a certain die is biased so that the probability of showing a six is less than $\frac { 1 } { 6 }$. He plans to throw the die 25 times and if it shows a six on fewer than 2 throws, he will conclude that the die is biased in this way.\\
(i) Find the probability of a Type I error and state the significance level of the test.
Joshi now decides to throw the die 100 times. It shows a six on 9 of these throws.\\
(ii) Calculate an approximate $95 \%$ confidence interval for the probability of showing a six on one throw of this die.
\hfill \mbox{\textit{CAIE S2 2012 Q3 [7]}}