| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find or state significance level |
| Difficulty | Standard +0.3 This is a straightforward application of binomial hypothesis testing with clear parameters. Part (i) requires a direct binomial probability calculation under H₀ (p=0.2, n=22, X≤1). Parts (ii) and (iii) test understanding of Type I/II error definitions—Type I probability equals significance level (no calculation needed), and Type II requires calculating P(X≥2) under the alternative p=0.09. All steps are routine with no conceptual surprises, making this slightly easier than average. |
| 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 |
|---|---|---|
| \(P(0, 1) = 0.8^{22} + 0.2 \times 0.8^{21} = 0.0480\) (4.8%) | M1, M1, A1 | For identifying the correct probability. For binomial probs with C and powers summing to 22. Correct answer accept 0.048. |
| (ii) \(P(\text{Type I error}) = 0.0480\) | B1ℜ | Ft on their (i). NB M1 M1 from (i) can be recovered in (ii) if not scored in (i). |
| (iii) \(P(\text{Type II error}) = 1 - P(0, 1) = 1 - (0.91^{22} + 0.09 \times 0.91^{21} ×_{22}C_1) = 0.601\) | M1, M1, A1 | Identifying the correct probability. Binomial probs with 0.09 and 0.91. Correct answer. |
**(i)** $X \sim B(22, 0.2)$
$P(0, 1) = 0.8^{22} + 0.2 \times 0.8^{21} = 0.0480$ (4.8%) | M1, M1, A1 | For identifying the correct probability. For binomial probs with C and powers summing to 22. Correct answer accept 0.048.
**(ii)** $P(\text{Type I error}) = 0.0480$ | B1ℜ | Ft on their (i). NB M1 M1 from (i) can be recovered in (ii) if not scored in (i).
**(iii)** $P(\text{Type II error}) = 1 - P(0, 1) = 1 - (0.91^{22} + 0.09 \times 0.91^{21} ×_{22}C_1) = 0.601$ | M1, M1, A1 | Identifying the correct probability. Binomial probs with 0.09 and 0.91. Correct answer.
---4 At a certain airport 20\% of people take longer than an hour to check in. A new computer system is installed, and it is claimed that this will reduce the time to check in. It is decided to accept the claim if, from a random sample of 22 people, the number taking longer than an hour to check in is either 0 or 1 .\\
(i) Calculate the significance level of the test.\\
(ii) State the probability that a Type I error occurs.\\
(iii) Calculate the probability that a Type II error occurs if the probability that a person takes longer than an hour to check in is now 0.09 .
\hfill \mbox{\textit{CAIE S2 2007 Q4 [7]}}