# Question 4:

## Part 4(a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: no association between amount spent and sex $H_1$: some association between amount spent and sex | B1 (both) | Hypotheses must be right way round, in context, and must not mention 'correlation' |
| **Total: 1** | | |

## Part 4(a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expected frequency $= 62 \times 102 \div 200 = 31.62$ | B1 | Do not allow 31.6 |
| Contribution $= (34 - 31.62)^2 / 31.62 = 0.1791$ | M1 A1 | For valid attempt at $\frac{(O-E)^2}{E}$; **NB answer given** |
| **Total: 3** | | |

# Question 4(a)(iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Refer to $X_4^2$; Critical value at 5% level = 9.488 | B1 | For 4 degrees of freedom |
| Critical value = 9.488 | B1 CAO | For cv |
| $3.205 < 9.488$ | M1 | |
| Result is not significant | A1 | For not significant; allow $p = 0.524$, $0.524 > 0.05$; conclusion must be stated to earn A1; allow 'do not reject $H_0$' and condone 'accept $H_0$' or 'reject $H_1$'; FT if cv consistent with their d.o.f. |
| There is insufficient evidence to suggest any association between amount spent and sex. | E1 | Dependent on previous A1 and final comment must be in context and not mention correlation; SC1 for correct final conclusion where previous A1 omitted but M1 awarded |
| | **5** | |

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# Question 4(b):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 400$ | B1 | Hypotheses in words must refer to population mean |
| $H_1: \mu < 400$ | B1 | |
| Where $\mu$ denotes the population mean (weight of the loaves) | B1 | For definition of $\mu$ |
| $\bar{x} = 396.5$ | B1 | For sample mean |
| Test statistic $= \dfrac{396.5 - 400}{5.7/\sqrt{6}} = \dfrac{-3.5}{2.327} = -1.504$ | M1, A1FT | M1 must include $\sqrt{6}$; use their sample mean; condone numerator reversed for M1 but award A1 only if test statistic of 1.504 is compared with a positive $z$-value |
| 5% level 1-tailed critical value of $z = -1.645$ | B1 | For $\pm 1.645$ |
| $-1.504 > -1.645$ so not significant | M1 | For sensible comparison leading to a conclusion; dependent on previous M1 |
| There is insufficient evidence to reject $H_0$ | | |
| There is insufficient evidence to suggest that the true mean weight of the loaves is lower than the minimum specified value of 400 grams. | A1 | For conclusion in context; FT their sample mean only if hypotheses are correct |
| | **9** | |

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# Additional Notes Q4(b):

| Method | Working | Marks |
|---|---|---|
| $\sigma$ estimated | Sample mean 7.079... used in place of 5.7 leads to test statistic $-1.212$... | M1A0, remaining marks available |
| Critical Value Method | $400 - 1.645 \times 5.7 \div \sqrt{6} = 396.17...$ | M1B1...A1 |
| | $400 + 1.645 \times 5.7 \div \sqrt{6}$ | M1B1A0 |
| | $396.5 > 396.2$ | M1 for sensible comparison (B1 for 396.5) |
| 90% CI Method | CI centred on 396.5; $\pm 1.645 \times 5.7 \div \sqrt{6}$ | M1 B1 |
| | $= (392.67, 400.33)$; contains 400 | A1, M1 |
| Probability Method | $P(\bar{x} < 396.5) = 0.0663$ | M1 A1 (and B1 for 396.5) |
| | $0.0663 > 0.05$ | M1 for sensible comparison |
| Two-tailed test | Max B1 B0 B1 B1 M1 A1 B1 (for $cv = -1.96$) M1 A0 | |