S4 June 2013 (R) Q6
6. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are
\[\hat{\mu} = 502 \qquad s^2 = 5.6\]Stating your hypotheses clearly, test at the 1% level of significance
(a) whether or not the mean amount of water in a bottle is more than 500 ml, (5)
(b) whether or not the standard deviation of the amount of water in a bottle is less than 3 ml. (5)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu = 500\) [accept \(\leqslant 500\)], \(\mathrm{H}_1 : \mu \gt 500\) | B1 |
| \(t = \dfrac{502 - 500}{\sqrt{5.6}/\sqrt{12}} = 2.93\) | M1A1 |
| critical value \(t_{11}(1\%) = 2.718\) | B1 |
| sufficient evidence that the mean amount of water is more than 500 ml | A1 ft |
| (5) |
Notes
M1 attempt at correct statistic
1st A1 awrt 2.93
2nd A1ft correct contextual comment including amount, water and 500
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2 = 9\) or \((\sigma = 3)\), \(\mathrm{H}_1 : \sigma^2 \lt 9\) or . \((\sigma \lt 3)\) | B1 |
| test statistic \(\dfrac{11s^2}{\sigma^2} =,\ \dfrac{61.6}{9} = 6.84\) | M1 A1 |
| critical values \(\chi^2_{11}(1\%)\) lower tail \(= 3.053\) | B1 |
| Insufficient evidence to suggest that the standard deviation of the amount of water is less than 3 | A1cso |
| (5) | |
| (10 marks) |
Notes
1st B1 Both hypotheses, must use \(\sigma\)
2nd B1 for critical value, this should be compatible with their alternative hypothesis
3rd A1cso cso. contextual comment, include standard deviation/ variance and water