## Question 4:

### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: no association between student smoking and parent smoking; $H_1$: some association between student smoking and parent smoking | B1 | Correct hypotheses in context; NB if $H_0$, $H_1$ reversed do not award first B1 or final B1dep*; Allow hypotheses in terms of independence; Do not allow "relationship" or "correlation" for "association" |
| Expected frequencies: Student smokes/Parent smokes = 15.2; Student smokes/Parent does not smoke = 32.8; Student does not smoke/Parent smokes = 22.8; Student does not smoke/Parent does not smoke = 49.2 | B1, B1 | For at least one row/column of expected values correct; All correct |
| Contributions: Student smokes/Parent smokes = 2.213; Student smokes/Parent does not smoke = 1.026; Student does not smoke/Parent smokes = 1.475; Student does not smoke/Parent does not smoke = 0.684 | M1, A1 | For valid attempt at $\frac{(O-E)^2}{E}$; All correct to 3 d.p.; NB these M1A1 marks cannot be implied by a correct final value of $X^2$ |
| $X^2 = 5.398$ | B1 | Allow awrt 5.40; Do not penalise use of Yates correction, giving $X^2 = 4.51$ |
| Refer to $\chi^2_1$; Critical value at 5% level $= 3.841$ | B1, B1 | For 1 degree of freedom; CAO for cv; $p$ value $= 0.02016$; No further marks from here if wrong or omitted, unless $p$-value used instead |
| Result is significant | B1* | For significant oe; FT their test statistic |
| There is sufficient evidence to **suggest** that there is **association between student smoking and parent smoking** | B1dep* | NB if $H_0$, $H_1$ reversed do not award first B1 or final B1dep*; For **non-assertive** conclusion in context; Do not allow "relationship" or "correlation" for "association" |

**[10 marks]**

### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = 88.2/100 = 0.882$ | B1 | For 0.882 seen |
| $s = \sqrt{\frac{78.68 - (88.2)^2/100}{99}} = \sqrt{\frac{0.8876}{99}} = \sqrt{0.0089657}$ | M1 | For correctly structured calculation for the sample standard deviation or variance |
| $= 0.0947$ (allow 0.095 www) | A1 | Allow A1 for $s^2 = 0.0089657$; or 0.00897 (allow 0.0090) |
| $H_0: \mu = 0.87$; $H_1: \mu > 0.87$; where $\mu$ denotes the **mean nicotine** content (of cigarettes of this brand in the population) | B1, B1 | For both correct; For definition of $\mu$ in context; Hypotheses in words only must refer to population; Do not allow other symbols unless clearly defined as population mean |
| Test statistic $= \frac{0.882 - 0.87}{0.0947/\sqrt{100}} = \frac{0.012}{0.00947} = 1.267$ | M1*, A1 | Including correct use of $\sqrt{100}$; CAO; FT their $s$ |
| Upper 1% level 1-tailed critical value of $z = 2.326$ | B1 | For 2.326 **No further marks from here if wrong** |
| $1.267 < 2.326$ (Not significant) | M1dep* | For sensible comparison leading to a conclusion |
| There is **insufficient evidence to suggest** that the **mean nicotine content** of this brand is **greater** than 0.87mg | A1 | For non-assertive conclusion in words and in context; FT only candidate's test statistic |

**[10 marks]**

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