# Question 10: Exemplars

## Hypotheses Section

### Exemplar A:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.034$ | B1 | Correct null hypothesis with parameter notation |
| $H_1: \mu \neq 0.034$ where $\mu$ = (pop) mean pollutant level | B1 | Correct two-tailed alternative with definition of $\mu$ |

### Exemplar B:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.034$ | B1 | Correct null hypothesis |
| $H_1: \mu \neq 0.034$ | B0 | Missing definition of $\mu$ |

### Exemplar C:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: The (pop) mean pollutant level is $0.034$ | B1 | Correct in words |
| $H_1$: The (pop) mean pollutant level is not $0.034$ | B0 | Must be in terms of parameter values — see Specimen paper mark scheme |

### Exemplar D:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0 = 0.034$, $H_0 \neq 0.034$ | B0B0 | No parameter used |

### Exemplar E:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.034$ | B1 | Correct null hypothesis |
| $H_1: \mu = 0.0325$ where $\mu$ = (pop) mean pollutant level | B0 | $H_1$ must be an inequality |

---

## Probability and Conclusion Section

### Exemplar F:
| Answer | Marks | Guidance |
|--------|-------|----------|
| No statement of distribution | — | |
| $P(\bar{X} = 0.0325) = 0.0486$ | M1A1 | Method and correct probability |
| $0.0486 > 0.025$ | A1 | Correct comparison |
| Don't reject $H_0$ | M1 | Correct decision |
| Likely that mean level of pollutant hasn't changed | A1 | Correct conclusion in context |

### Exemplar G:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\bar{X} = 0.0325) = 0.0486$ | M1A1 | |
| $0.0486 > 0.025$ | A1 | |
| Accept $H_0$ | M1 | |
| There is evidence that mean level of pollutant hasn't changed | A0 | Conclusion not stated correctly |

### Exemplar H:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\bar{X} < 0.0325) = 0.951$ | M1A0 | Wrong probability used |
| $0.951 > 0.025$ | A0 | |
| Insufficient evidence that poll't level has changed | M0A0 | |

### Exemplar I:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\bar{X} > 0.0325) = 0.951$ | M1A1 | |
| $0.951 > 0.025$ | A0 | |
| Sufficient evidence that mean poll't level has changed | M0A0 | Incorrect conclusion |

### Exemplar J:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{X} \sim N(0.034, 0.000000818)$ | — | |
| $P(\bar{X} < 0.0325) = 0.013$ | M1A0 | Incorrect variance used |
| $0.013 < 0.025$ | A0 | |
| Sufficient evidence that level has changed | M1A1 | Follow through conclusion |

### Exemplar K:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mu \pm 1.96\sigma = 0.0322$ to $0.0358$ | M1A1 | BOD |
| $0.0325$ lies within this range | A1 | |
| Reject $H_1$ | M1 | |
| Insufficient evidence that level of poll't has decreased | A0 | |

### Exemplar L:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $CV = 0.0322$ | M1A1 | |
| $0.0325 > 0.0322$ | A1 | |
| Reject $H_0$. Evidence that level of poll't has changed. | M0A0 | Incorrect decision leads to wrong conclusion |

### Exemplar M:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0.0322 - 0.034) \div \sqrt{0.0000409/50} = -1.66$ | M1A1 | |
| $1.66 < 1.96$ | A1 | |
| Don't reject $H_0$. Level of poll't hasn't changed. | M1A0 | Conclusion lacks context detail |

---

## 1-tail Hypotheses

### Exemplar N:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.034$ | B1 | |
| $H_1: \mu < 0.034$ where $\mu$ = (pop) mean pollutant level | B0 | Wrong tail |

### Exemplar O:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 0.034$ | B0 | |
| $H_1: \mu < 0.034$ | B0 | Wrong tail and no definition |

### Exemplar P:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: The (pop) mean pollutant level is $0.034$ | B0 | |
| $H_1$: The (pop) mean pollutant level is less than $0.034$ | B0 | Wrong tail |