| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Moderate -0.3 This is a straightforward one-tailed binomial hypothesis test with clear parameters (n=10, p=0.4, significance level 5%). The calculation is simple (finding P(X≤1) under H₀), and part (ii) requires only basic interpretation of hypothesis test conclusions. Slightly easier than average due to small sample size making calculations trivial and standard test structure. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B1B1 | |
| \(N(4, 2.4)\) | M1 | |
| \(P(\leq 1) = 0.0533\) | A0 | |
| \(> 0.05\) | A0 | |
| So do not reject \(H_0\). Insufficient evidence that % who book with travel agents reduced | M0 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B1B1 | |
| \(B(10, 0.4)\) | M1 | |
| \(P(= 1) = 0.0464\) | A1 | *[allow this]* |
| \(\leq 0.05\) | A1 | |
| So reject \(H_0\) | M1 | |
| Insufficient evidence that % who book with travel agents reduced | A0 | 6 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B1B1 | |
| \(B(10, 0.4)\) | M1 | |
| \(P(= 1) = 0.0404\) | A0 | *[look out for this]* |
| \(< 0.05\) so reject \(H_0\) | A0 | |
| Significant evidence that % who book with travel agents reduced | M0, A0 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B1B1 | |
| \(B(10, 0.4)\) | M1 | |
| \(P(\geq 1) = 0.9939\) | A0 | |
| \(> 0.95\) | A0 | |
| So reject \(H_0\) | M0 | |
| Insufficient evidence that % who book with travel agents reduced | A0 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p \neq 0.4\) | B1B0 | *[two-tailed]* |
| \(B(10, 0.4)\) | M1 | |
| \(P(= 1) = 0.0464\) | A1 | |
| \(> 0.025\) | A0 | |
| So do not reject \(H_0\) | M1 | |
| Insufficient evidence that % who book with travel agents reduced | A1 | 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B1B1 | |
| \(B(10, 0.4)\) | M1 | |
| \(P(= 1) = 0.0464\) | A1 | |
| *[no explicit comparison]* | A0 | |
| So reject \(H_0\). Significant evidence that % who book with travel agents reduced | M1, A1 | 6 marks total |
# Question 5(i): Specific Examples
## Case α (H₁: p < 0.4, N(4, 2.4))
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B1B1 | |
| $N(4, 2.4)$ | M1 | |
| $P(\leq 1) = 0.0533$ | A0 | |
| $> 0.05$ | A0 | |
| So do not reject $H_0$. Insufficient evidence that % who book with travel agents reduced | M0 | **3 marks total** |
## Case β (H₁: p < 0.4, B(10, 0.4))
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B1B1 | |
| $B(10, 0.4)$ | M1 | |
| $P(= 1) = 0.0464$ | A1 | *[allow this]* |
| $\leq 0.05$ | A1 | |
| So reject $H_0$ | M1 | |
| Insufficient evidence that % who book with travel agents reduced | A0 | **6 marks total** |
## Case γ (H₁: p < 0.4, B(10, 0.4))
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B1B1 | |
| $B(10, 0.4)$ | M1 | |
| $P(= 1) = 0.0404$ | A0 | *[look out for this]* |
| $< 0.05$ so reject $H_0$ | A0 | |
| Significant evidence that % who book with travel agents reduced | M0, A0 | **3 marks total** |
## Case δ (H₁: p < 0.4, B(10, 0.4))
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B1B1 | |
| $B(10, 0.4)$ | M1 | |
| $P(\geq 1) = 0.9939$ | A0 | |
| $> 0.95$ | A0 | |
| So reject $H_0$ | M0 | |
| Insufficient evidence that % who book with travel agents reduced | A0 | **3 marks total** |
## Case ε (Two-tailed, B(10, 0.4))
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p \neq 0.4$ | B1B0 | *[two-tailed]* |
| $B(10, 0.4)$ | M1 | |
| $P(= 1) = 0.0464$ | A1 | |
| $> 0.025$ | A0 | |
| So do not reject $H_0$ | M1 | |
| Insufficient evidence that % who book with travel agents reduced | A1 | **5 marks total** |
## Case ζ (H₁: p < 0.4, B(10, 0.4), no explicit comparison)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B1B1 | |
| $B(10, 0.4)$ | M1 | |
| $P(= 1) = 0.0464$ | A1 | |
| *[no explicit comparison]* | A0 | |
| So reject $H_0$. Significant evidence that % who book with travel agents reduced | M1, A1 | **6 marks total** |
---5 A travel company finds from its records that $40 \%$ of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.\\
(i) Test at the $5 \%$ significance level whether the percentage of customers who book with travel agents has decreased.\\
(ii) The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.
\hfill \mbox{\textit{OCR S2 2011 Q5 [8]}}