| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Standard +0.3 This is a straightforward application of binomial hypothesis testing with standard procedures: finding a critical region from tables, comparing a test statistic, and calculating Type II error probability. While it requires multiple steps and understanding of hypothesis testing concepts, all techniques are routine S2 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)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 |
|---|---|---|
| (i) \(B(14, 0.7)\) | M1 | \(B(14, 0.7)\) stated or implied, e.g. N(9.8, 2.94), can be recovered |
| \(CR \text{ is } \ge 13\) | A1 | CV 13, or \(\{12, 13, 14\}\), allow = but not inequalities |
| with probability 0.0475 | A1 | 3 |
| 3 | ||
| (ii) \(H_0: p = 0.7, H_1: p > 0.7\) | B2 | Both, B2. Allow \(\pi\). One error, B1, but r, \(x\) etc: B0 |
| \(12 < 13\) | B1 | Compare CV from correct null and inequality with 12, |
| or P\((\ge 12) = 0.1608\) and \(> 0.05\) or P\((\le 12) = 0.8392\) and \(< 0.95\) | ||
| M1 FT 5 | Withhold if inconsistent | |
| Contextualised, acknowledge uncertainty | ||
| [SR: Normal or Po: (i) M1, (ii) B2 maximum] [0.9932 or 0.0068 probably B2 maximum] [0.9932 or 0.0068 probably B2 maximum] | ||
| (iii) \(B(14, 0.8)\) | M1 | \(B(14, 0.8)\) stated or implied, allow from B(14, 0.75) |
| \(P(\le 12)\) from B(14, 0.8) | M1 | Attempt prob of acceptance region, e.g. 0.8990, √on (i) |
| \(0.8021\) | A1 | 3 |
**(i)** $B(14, 0.7)$ | M1 | $B(14, 0.7)$ stated or implied, e.g. N(9.8, 2.94), can be recovered
$CR \text{ is } \ge 13$ | A1 | CV 13, or $\{12, 13, 14\}$, allow = but not inequalities
with probability 0.0475 | A1 | 3 | Exactly correct CR, and supporting prob .[0475 or .9525 seen
| | 3
**(ii)** $H_0: p = 0.7, H_1: p > 0.7$ | B2 | Both, B2. Allow $\pi$. One error, B1, but r, $x$ etc: B0
$12 < 13$ | B1 | Compare CV from correct null and inequality with 12,
or P$(\ge 12) = 0.1608$ and $> 0.05$ or P$(\le 12) = 0.8392$ and $< 0.95$ | | | Correct method & conclusion, requires like-with-like; CV method needs $\geq 13$ or $< 12$; $p$ method needs $\geq 12$ or $< 12$
| M1 FT 5 | Withhold if inconsistent
| | | Contextualised, acknowledge uncertainty
| | | [SR: Normal or Po: (i) M1, (ii) B2 maximum] [0.9932 or 0.0068 probably B2 maximum] [0.9932 or 0.0068 probably B2 maximum]
**(iii)** $B(14, 0.8)$ | M1 | $B(14, 0.8)$ stated or implied, allow from B(14, 0.75)
$P(\le 12)$ from B(14, 0.8) | M1 | Attempt prob of acceptance region, e.g. 0.8990, √on (i)
$0.8021$ | A1 | 3 | Answer 0.802 or a.r.t. 0.80219 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion $p$ of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the $5 \%$ significance level, on a random sample of 14 patients who suffer from the disease.\\
(i) Find the critical region for the test.\\
(ii) Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.\\
(iii) Find the probability that the test results in a Type II error if in fact $p = 0.8$.
RECOGNISING ACHIEVEMENT
\hfill \mbox{\textit{OCR S2 2011 Q9 [11]}}