S4 June 2005 Q3
3. A machine is set to fill bags with flour such that the mean weight is 1010 grams.
To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows.
1010 1015 1005 1000 998 1008 1012 1007
Carry out a suitable test, at the 5% significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.) (8)
| Scheme | Marks |
|---|---|
| Let \(x\) represent weight of flour | |
| \(\sum x = 8055 \quad \therefore\ \bar{x} = 1006.875\) awrt 1006.9 | B1 |
| \(\sum x^2 = 8110611 \quad \therefore\ s^2 = \dfrac{1}{7}\left\{8110611 - \dfrac{8055^2}{8}\right\} = 33.26785\ldots\) awrt 33.3 | M1 |
| \(\therefore\ s = 5.767828\ldots\) or awrt 5.77 (Allow from calculator) | A1 |
| \(\mathrm{H}_0 : \mu = 1010\,;\quad \mathrm{H}_1 : \mu \lt 1010\) both | B1 |
| CV: \(|t| = 1.895\) | B1 |
| \(t = \left|\dfrac{1006.875 - 1010}{5.767828\ldots/\sqrt{8}}\right| = \pm 1.5324\) Using \(\dfrac{\bar{x} - \mu}{s/\sqrt{n}}\) | M1 |
| awrt \(-1.53\) | A1 |
| Since \(-1.53\) is not in the critical region \((t \lt -1.895)\) there is insufficient evidence to reject \(\mathrm{H}_0\) and thus consistent with the mean weight of flour delivered by the machine is 1010 g. | A1ft |
| (8) |
Notes
(corrected from the printed mark scheme: the scheme prints “awrt 33.7” for \(s^2\); the value it shows is \(33.26785\ldots\), so awrt 33.3)