S4 June 2014 Q1
1. A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml.
The amount of oil, \(x\) ml, in each of 8 randomly selected bottles is recorded, and the following statistics are obtained.
\[\bar{x} = 92.875 \qquad s = 8.3055\]Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml.
Stating your hypotheses clearly, test, at the 5% significance level, whether or not Malcolm’s belief is supported. (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 100 \quad \mathrm{H}_1 : \mu \lt 100\) | B1 |
| \(t = \dfrac{|\bar{x} - \mu|}{s/\sqrt{n}} = \dfrac{|92.875 - 100|}{8.3055/\sqrt{8}} = 2.4264\ldots\) or \(\dfrac{c - 100}{8.3055/\sqrt{8}} = -1.895\ \therefore\ \text{CR } c \lt 94.435\) | M1A1 |
| \(t_7(5\%) = \pm 1.895\) | B1 |
| There is evidence to reject \(\mathrm{H}_0\). Malcolm’s belief is supported or there is evidence that the amount of oil placed in bottles is less than 100ml | A1ft |
| (5) |
Notes
B1 both hypotheses
M1 either \(\dfrac{|92.875 - 100|}{8.3055/\sqrt{8}}\) or \(p = 0.0228\) or \(\dfrac{c - 100}{8.3055/\sqrt{8}} = -(\text{a } t \text{ value})\)
A1 awrt 2.43 or awrt 94.4 or awrt 0.0228
B1 \(\pm 1.895\) or \(0.0228 \lt 0.05\) (must have correct comparison for hypotheses)
A1ft Do Not allow contradictions
(Corrected from the printed mark scheme: the conclusion printed “100mm” for 100 ml.)