| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Moderate -0.5 This is a straightforward application of Type I and Type II error definitions with binomial probability calculations. Part (i) requires calculating P(X ≥ 9 or X ≤ 1) under H₀: p=0.5, which is routine. Part (ii) requires the same calculation under p=0.7. Both parts involve standard binomial probability computation with no conceptual challenges beyond knowing the definitions. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{Type I error}) = 0.0215\) AG | M1, M1, M1, A1 (4 marks) | For \(P(9\) or \(10H)\); For \(P(9\) or \(10T)\); For identifying outcome for Type I error; For obtaining given answer legitimately |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 0.851\) | M1, M1, M1, A1 (4 marks) | For evaluating \(P(9\) or \(10H)\) with \(P(H) = 0.7\); For evaluating \(P(9\) or \(10T)\) with \(P(T) = 0.3\); For identifying outcome for Type II error; For correct answer (SR 0.851 no working B2) |
**(i)** $P(9 \text{ or } 10H) = (0.5)^x(0.5) \times_{10}C_9 + (0.5)^{10}$
$= 0.01074$
$P(9 \text{ or } 10T) = 0.01074$
$P(\text{Type I error}) = 0.0215$ AG | M1, M1, M1, A1 (4 marks) | For $P(9$ or $10H)$; For $P(9$ or $10T)$; For identifying outcome for Type I error; For obtaining given answer legitimately
**(ii)** $P(9 \text{ or } 10H) = (0.7)^9 \times (0.3) \times_{10}C_9 + (0.7)^{10}$
$= 0.1493$
$P(9 \text{ or } 10T) = (0.3)^{10} \times (0.7) \times_{10}C_9 + (0.3)^{10}$
$= 0.000143$
$P(\text{type II error}) = 1 - 0.1493 - 0.000143$
$= 0.851$ | M1, M1, M1, A1 (4 marks) | For evaluating $P(9$ or $10H)$ with $P(H) = 0.7$; For evaluating $P(9$ or $10T)$ with $P(T) = 0.3$; For identifying outcome for Type II error; For correct answer (SR 0.851 no working B2)
---5 To test whether a coin is biased or not, it is tossed 10 times. The coin will be considered biased if there are 9 or 10 heads, or 9 or 10 tails.\\
(i) Show that the probability of making a Type I error in this test is approximately 0.0215 .\\
(ii) Find the probability of making a Type II error in this test when the probability of a head is actually 0.7.
\hfill \mbox{\textit{CAIE S2 2002 Q5 [8]}}