3 Pitted black olives in brine are sold in jars labelled " 340 grams net weight". Two machines, A and B, independently fill these jars with olives before the brine is added. The weight, \(X\) grams, of olives delivered by machine A may be modelled by a normal distribution with mean \(\mu _ { X }\) and standard deviation 4.5. The weight, \(Y\) grams, of olives delivered by machine B may be modelled by a normal distribution with mean \(\mu _ { Y }\) and standard deviation 5.7. The mean weight of olives from a random sample of 10 jars filled by machine A is found to be 157 grams, whereas that from a random sample of 15 jars filled by machine \(B\) is found to be 162 grams. Test, at the \(1 \%\) level of significance, the hypothesis that \(\mu _ { X } = \mu _ { Y }\).
(6 marks)

| **Main Solution** |
|---|
| $H_0: \mu_x = \mu_y$; $H_1: \mu_x \neq \mu_y$ | B1 | Both |
| $\text{SL } \alpha = 0.01(1\%)$; $\text{CV } z = (\pm) 2.57 \text{ to } 2.58$ | B1 | AWFW (2.5758) |
| $z = \frac{\|157 - 162\|}{\sqrt{\frac{4.5^2}{10} + \frac{5.7^2}{15}}} = $ | M1 | Numerator |
| | M1 | Denominator |
| $(\pm) 2.44 \text{ to } 2.445$ | A1 | AWFW (2.4424) |
| No evidence, at 1% level, to reject hypothesis that $\mu_x = \mu_y$ | A1↑ | ft on $z$, CV and signs; or equivalent; 6 marks |

**Total: 6 marks**

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3 Pitted black olives in brine are sold in jars labelled " 340 grams net weight". Two machines, A and B, independently fill these jars with olives before the brine is added.

The weight, $X$ grams, of olives delivered by machine A may be modelled by a normal distribution with mean $\mu _ { X }$ and standard deviation 4.5.

The weight, $Y$ grams, of olives delivered by machine B may be modelled by a normal distribution with mean $\mu _ { Y }$ and standard deviation 5.7.

The mean weight of olives from a random sample of 10 jars filled by machine A is found to be 157 grams, whereas that from a random sample of 15 jars filled by machine $B$ is found to be 162 grams.

Test, at the $1 \%$ level of significance, the hypothesis that $\mu _ { X } = \mu _ { Y }$.\\
(6 marks)

\hfill \mbox{\textit{AQA S3 2008 Q3 [6]}}