| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State hypotheses with additional parts |
| Difficulty | Moderate -0.8 This is a straightforward hypothesis testing question requiring standard recall of null/alternative hypotheses format, interpretation of a p-value against a significance level, and understanding that B(150,p) means 150 trials. All three parts are routine applications of basic statistical concepts with no problem-solving or novel insight required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p = \frac{1}{3}\) \(\quad\) \(H_1: p < \frac{1}{3}\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.0084 < 0.01\) | B1 | Allow \(P(N \leqslant 36) < 0.01\) or 1% |
| There is evidence that \(p\) has decreased | B1 dep | Allow '\(p\) has decreased' or \(p < \frac{1}{3}\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 150 | B1 | |
| Total: 1 |
## Question 2:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = \frac{1}{3}$ $\quad$ $H_1: p < \frac{1}{3}$ | B1 | |
| **Total: 1** | | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.0084 < 0.01$ | B1 | Allow $P(N \leqslant 36) < 0.01$ or 1% |
| There is evidence that $p$ has decreased | B1 dep | Allow '$p$ has decreased' or $p < \frac{1}{3}$ |
| **Total: 2** | | |
**Part (iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| 150 | B1 | |
| **Total: 1** | | |
---2 A headteacher models the number of children who bring a 'healthy' packed lunch to school on any day by the distribution $\mathrm { B } ( 150 , p )$. In the past, she has found that $p = \frac { 1 } { 3 }$. Following the opening of a fast food outlet near the school, she wishes to test, at the $1 \%$ significance level, whether the value of $p$ has decreased.\\
(i) State the null and alternative hypotheses for this test.\\
On a randomly chosen day she notes the number, $N$, of children who bring a 'healthy' packed lunch to school. She finds that $N = 36$ and then, assuming that the null hypothesis is true, she calculates that $\mathrm { P } ( N \leqslant 36 ) = 0.0084$.\\
(ii) State, with a reason, the conclusion that the headteacher should draw from the test.\\
(iii) According to the model, what is the largest number of children who might bring a packed lunch to school?\\
\hfill \mbox{\textit{CAIE S2 2018 Q2 [4]}}