2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable \(B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)\). The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable \(G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)\). The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the \(5 \%\) level of significance, the hypothesis that \(\mu _ { B } = \mu _ { G }\).
[0pt] [6 marks]

# Question 2:

| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu_B = \mu_G$, $H_1: \mu_B \neq \mu_G$ | B1 | Both hypotheses correct |
| $\bar{B} - \bar{G} = 21.35 - 21.90 = -0.45$ | M1 | Correct difference of means |
| $\text{Var}(\bar{B} - \bar{G}) = \frac{0.5625}{20} + \frac{0.9025}{15}$ | M1 | Correct variance formula for difference of sample means |
| $= 0.028125 + 0.060167 = 0.088292$ | A1 | Correct variance value |
| $z = \frac{-0.45}{\sqrt{0.088292}} = \frac{-0.45}{0.29714} = -1.514$ | M1 | Correct test statistic calculation |
| Critical value $z = \pm 1.96$ (5%, two-tailed) | B1 | Correct critical value stated |
| $|-1.514| < 1.96$, do not reject $H_0$ | A1 | Correct conclusion — insufficient evidence that $\mu_B \neq \mu_G$ |

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2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly.

The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable $B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)$.

The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable $G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)$.

The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg .

Test, at the $5 \%$ level of significance, the hypothesis that $\mu _ { B } = \mu _ { G }$.\\[0pt]
[6 marks]

\hfill \mbox{\textit{AQA S3 2014 Q2 [6]}}