S4 June 2015 Q2
2. Fred is a new employee in a delicatessen. He is asked to cut cheese into 100 g blocks. A random sample of 8 of these blocks of cheese is selected. The weight, in grams, of each block of cheese is given below
94, 106, 115, 98, 111, 104, 113, 102
Given that the weights of the blocks of cheese are independent,
The delicatessen manager expects the standard deviation of the weights of the blocks of cheese cut by an employee to be less than 5 g. Any employee who does not achieve this target is given training.
A second employee, Olga, has just been given training. Olga is asked to cut cheese into 100 g blocks. A random sample of 20 of these blocks of cheese is selected. The weight of each block of cheese, \(x\) grams, is recorded and the results are summarised below.
\[\bar{x} = 102.6 \qquad s^2 = 19.4\]Given that the assumption in part (b) is also valid in this case,
| Scheme | Marks |
|---|---|
| \(n = 8 \quad \sum x = 843 \quad \sum x^2 = 89211\) \(\therefore\ \bar{x} = 105.375\) \(s^2 = \dfrac{8}{7}\left(\dfrac{89211}{8} - 105.375^2\right) = 54.2678\ldots\) or \(s^2 = \dfrac{1}{7}\left(89211 - \dfrac{843^2}{8}\right) = 54.2678\ldots\) | M1A1 |
| Confidence interval is given by \(\dfrac{7 \times 54.267\ldots}{14.067} \lt \sigma^2 \lt \dfrac{7 \times 54.267\ldots}{2.167}\) | M1B1 |
| \(\therefore\ 27.004\ldots \lt \sigma^2 \lt 175.299\ldots\) | |
| \(5.1966\ldots \lt \sigma \lt 13.240\ldots\) | M1d A1 |
| (6) |
Notes
1st M1 attempting \(s\) or \(s^2\) 1st A1 awrt 54.3
2nd M1 for \(\dfrac{7s^2}{\chi^2}\)
B1 14.067 & 2.167
3rd M1d Dept on previous M mark. Rearranging leading to interval for σ- must square root
A1 awrt 5.20 and 13.2 (allow 5.2)
NB a correct interval gains full marks
| Scheme | Marks |
|---|---|
| Need to assume underlying Normal distribution for weights of blocks of cheese. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Lower limit of CI is \(\gt 5\ g\) suggests that Fred needs training. | B1ft |
| (1) |
Notes
B1ft on their CI must have Fred/He/employee (do not allow empoloyees) and training. They must have an interval in part(a)
| Scheme | Marks |
|---|---|
| To test \(\mathrm{H}_0 : \mu = 100,\ \ \mathrm{H}_1 : \mu \ne 100 \ \ (\mu \gt 100)\) where \(\mu\) is the mean weight of blocks of cheese | B1 |
| Test statistic \(t = \dfrac{102.6 - 100}{\sqrt{\frac{19.4}{20}}} = 2.6399\ldots\) | M1A1 |
| Critical value(s): \(t_{19} = (\pm)1.729\ (1.328)\) | B1 |
| In critical region, therefore significant evidence to reject \(\mathrm{H}_0\) and accept \(\mathrm{H}_1\) | A1ft |
| Significant evidence that the mean weight of the blocks of cheese is not 100 g (more than 100g) | B1cso |
| (6) | |
| (14 marks) |
Notes
1st B1 Both hypotheses with \(\mu\). Allow one-tail
1st M1 \(\dfrac{102.6 - 100}{\frac{s \text{ or } s^2}{\sqrt{20}}}\) 1st A1 awrt 2.64
2nd B1 allow \(p\) value of 0.0161 in place of critical value. CV must follow from H1
2nd A1ft a correct statement – do not allow contradicting non context statement.
3rd B1cso need correct conclusion in context containing the words in bold from a fully correct solution. For one tail need “more than 100g”