S4 June 2018 Q1
1. A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by \(\mathrm{N}(\mu, \sigma^2)\). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for \(\mu\) and \(\sigma^2\) are
\[\bar{x} = 202 \qquad \text{and} \qquad s^2 = 3.6\]Stating your hypotheses clearly, test, at the 1% level of significance, whether or not the mean weight of almonds in a packet is more than 200 g. (5)
| Scheme | Marks |
|---|---|
| To test \(\mathrm{H}_0 : \mu = 200, \quad \mathrm{H}_1 : \mu \gt 200\) | B1 |
| Test statistic \(t = \dfrac{202 - 200}{\sqrt{\frac{3.6}{10}}} = \dfrac{10}{3}\) or \(3.3333\ldots\) | M1A1 |
| Critical value(s): \(t_9 = (\pm)2.821\) | B1 |
| In critical region, therefore significant evidence to reject \(\mathrm{H}_0\) and accept \(\mathrm{H}_1\) | |
| Significant evidence that the mean weight of the packets of almonds is more than 200 g | A1ft |
| (5) | |
| (5 marks) |
Notes
1st B1 Both hypotheses with \(\mu\).
1st M1 Allow \(\pm\dfrac{202 - 200}{\frac{s}{\sqrt{10}}}\)
1st A1 awrt 3.33
2nd B1 allow \(p\) value of awrt 0.00438 in place of critical value. CV must follow from \(\mathrm{H}_1\), sign must match \(t\)-value or be \(\pm\)
2nd A1ft ft \(t\)-value if awarded B marks. Need correct conclusion in context containing the words mean weight, almonds or packets and 200g