| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (p < 0.19), requiring calculation of P(X ≤ 1) where X ~ B(20, 0.19) and comparison to 10% significance level. Part (ii) requires understanding that letters in words aren't independent. Slightly easier than average due to small n allowing exact calculation and explicit guidance on the test direction. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Horizontal straight line | B1 | |
| Positive parabola, symmetric about 0 | B1 | |
| Completely correct, including correct relationship between two; Don't need vertical lines or horizontal lines outside range, but don't give last B1 if horizontal line continues past "±1" | B1 | 3 |
| (ii) \(S\) is equally likely to take any value in range, \(T\) is more likely at extremities | B2 | Correct statement about distributions (not graphs) [Partial statement, or correct description for one only: B1] |
| (iii) \(\int_{-1}^{1} x^2 dx = \left[\frac{x^3}{3}\right]\) | M1 | Integrate f(x) with limits (−1, 1) or (t, 1) [recoverable if \(t\) used later] |
| \(\frac{1}{2}(1 - t^2) = 0.2\) or \(\frac{1}{2}(t^3 + 1) = 0.8\); \(t^3 = 0.6\); \(t = 0.8434\) | B1 | Correct indefinite integral |
| M1 | Equate to 0.2, or 0.8 if [−1, t] used | |
| M1 | Solve cubic equation to find \(t\) | |
| A1 | Answer, in range [0.843, 0.844] | 5 |
(i) Horizontal straight line | B1 |
Positive parabola, symmetric about 0 | B1 |
Completely correct, including correct relationship between two; Don't need vertical lines or horizontal lines outside range, but don't give last B1 if horizontal line continues past "±1" | B1 | 3
(ii) $S$ is equally likely to take any value in range, $T$ is more likely at extremities | B2 | Correct statement about distributions (not graphs) [Partial statement, or correct description for one only: B1]
(iii) $\int_{-1}^{1} x^2 dx = \left[\frac{x^3}{3}\right]$ | M1 | Integrate f(x) with limits (−1, 1) or (t, 1) [recoverable if $t$ used later]
$\frac{1}{2}(1 - t^2) = 0.2$ or $\frac{1}{2}(t^3 + 1) = 0.8$; $t^3 = 0.6$; $t = 0.8434$ | B1 | Correct indefinite integral
| M1 | Equate to 0.2, or 0.8 if [−1, t] used
| M1 | Solve cubic equation to find $t$
| A1 | Answer, in range [0.843, 0.844] | 56 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter $e$ appears $19 \%$ of the time. A certain encoded message of 20 letters contains one letter $e$.\\
(i) Using an exact binomial distribution, test at the $10 \%$ significance level whether there is evidence that the proportion of the letter $e$ in the language from which this message is a sample is less than in German, i.e., less than $19 \%$.\\
(ii) Give a reason why a binomial distribution might not be an appropriate model in this context.
\hfill \mbox{\textit{OCR S2 2007 Q6 [9]}}